tag:blogger.com,1999:blog-267065642016-12-09T09:51:01.428-05:00Thoughts On EconomicsRobert Vienneauhttp://www.blogger.com/profile/14748118392842775431noreply@blogger.comBlogger1004125tag:blogger.com,1999:blog-26706564.post-1151835560707333482016-12-31T03:00:00.000-05:002015-01-06T06:09:59.760-05:00WelcomeI study economics as a hobby. My interests lie in Post Keynesianism, (Old) Institutionalism, and related paradigms. These seem to me to be approaches for understanding actually existing economies.<br /><br />The emphasis on this blog, however, is mainly critical of neoclassical and mainstream economics. I have been alternating numerical counter-examples with less mathematical posts. In any case, I have been documenting demonstrations of errors in mainstream economics. My chief inspiration here is the Cambridge-Italian economist Piero Sraffa.<br /><br />In general, this blog is abstract, and I think I steer clear of commenting on practical politics of the day.<br /><br />I've also started posting recipes for my own purposes. When I just follow a recipe in a cookbook, I'll only post a reminder that I like the recipe.<br /><br /><B>Comments Policy:</B> I'm quite lax on enforcing any comments policy. I prefer those who post as anonymous (that is, without logging in) to sign their posts at least with a pseudonym. This will make conversations easier to conduct.Robert Vienneauhttp://www.blogger.com/profile/14748118392842775431noreply@blogger.com64tag:blogger.com,1999:blog-26706564.post-14417873967862035192016-12-09T07:23:00.000-05:002016-12-09T07:23:17.137-05:00Basic Commodities and Multiple Interest Rate AnalysisI have a new <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2882531">working paper</A> on the Social Science Research Network: <BLOCKQUOTE><B>Abstract:</B> This paper considers the application of multiple interest rate analysis to a model of the production of commodities by means of commodities. A polynomial, for the characteristic equation of the augmented input-output matrix, is used in defining the rate of profits in such a model. Only one root is found to be economically meaningful. No non-trivial application of multiple interest rate analysis is found in the analysis of the choice of technique. On the other hand, multiple interest rate analysis can be used in defining Net Present Value in an approximate model, in which techniques are represented as finite series of dated labor inputs. The product of the quantity of the first labor input and the composite interest rate approaches, in the limit, the difference between the labor commanded by and the labor embodied in final output in the full model. </BLOCKQUOTE><P>I am proud of some observations in this paper. Nevertheless, I think it tries to go in too many directions at once. It is also longer than I like. It may seem, at first glance, to be longer than it is. I have ten graphs scattered throughout. </P><P>Michael Osborne cannot deny that I have taken his research seriously. He needs somebody with more academic credibility than me to write on his topic, though. </P><P>This is one paper where I would mind being shown to be wrong. I did not find any use for more than one eigenvalue of what I am calling the augmented input-output matrix. If somebody can find something useful, along the line of multiple interest rate analysis, to say about all eigenvalues, I would be interested to hear of it. </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-46088481913425043232016-12-06T10:02:00.000-05:002016-12-06T10:02:09.479-05:00Bifurcations In Multiple Interest Rate Analysis<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><a href="https://4.bp.blogspot.com/-qYzQv2Y7oKc/WEbNxSgu2xI/AAAAAAAAAsE/HYrjB-ff6PYoPEySBlwuTZ23tLkNi8VngCLcB/s1600/ThreePolynomials.jpg" imageanchor="1" ><img border="0" src="https://4.bp.blogspot.com/-qYzQv2Y7oKc/WEbNxSgu2xI/AAAAAAAAAsE/HYrjB-ff6PYoPEySBlwuTZ23tLkNi8VngCLcB/s320/ThreePolynomials.jpg" width="320" height="240" /></a></td></tr><tr><td align="center"><b>Figure 1: Three Trinomials</b></td></tr></tbody></table><B>1.0 Introduction</B><P>Typically, in calculating the Internal Rate of Return (IRR), a polynomial function arises. The IRR is the smallest, non-negative rate of profits, as calculated from a root of this function. The other roots are almost always ignored as having no economic meaning. </P><P>Michael Osborne, as I understand it, is pursuing a research project of investigating the use of all the roots of such polynomial functions that arise in financial analysis. A polynomial of degree <I>n</I> has <I>n</I> roots in the complex plane. I have noticed that the roots, other than the IRR, for examples that might arise in practice, can vary in whether they are real, repeating, or complex. </P><P>Bifurcation analysis, as developed for the study of dynamic systems might therefore have an application in multiple interest rate analysis. (This post is not about a dynamic system. I do not know how many of these <A HREF="https://www.google.com/?gws_rd=ssl#q=bifurcation+%22theory+of+equations%22">results</A> are about the theory of equations, independently of dynamical systems.) On the other hand, Osborne typically presents his analyses in terms of complex numbers. So I am not sure that he need care about these details. </P><B>2.0 An Example</B><P>Table 1 specifies the technology to be analyzed in this post. This technology produces an output of corn at the end of one specified year. The production of corn requires inputs of flows of labor in each of the three preceding years (and no other inputs). The labor inputs, per unit corn output, are listed in the table. </P><TABLE BORDER="1" ALIGN="center"><CAPTION><B>Table 1: The Technology</B></CAPTION><TR><TD ALIGN="center"><B>Year<BR>Before<BR>Output</B></TD><TD ALIGN="center" COLSPAN="2"><B>Labor Hired<BR>for Each Technique</B></TD></TR><TR><TD ALIGN="center"><B>1</B></TD><TD ALIGN="center"><I>L</I><SUB>1</SUB> = 0.18 Person-Years</TD></TR><TR><TD ALIGN="center"><B>2</B></TD><TD ALIGN="center"><I>L</I><SUB>2</SUB> = 4.468 Person-Years</TD></TR><TR><TD ALIGN="center"><B>3</B></TD><TD ALIGN="center"><I>L</I><SUB>3</SUB> = 0.527438298 Person-Years</TD></TR></TABLE><P>Let a unit of corn be the numeraire. Suppose firms face a wage of <I>w</I> and a rate of profits, <I>r</I>, to be used for time discounting. Wages are assumed to be advanced. That is, workers are paid at the start of the year for each year in which they supply flows of labor. Accumulate all costs to the end of the year in which the harvest occurs. Then the Net Present Value for this technology is: </P><BLOCKQUOTE>NPV(<I>r</I>) = 1 - <I>w</I>[<I>L</I><SUB>1</SUB>(1 + <I>r</I>) + <I>L</I><SUB>2</SUB>(1 + <I>r</I>)<SUP>2</SUP> + <I>L</I><SUB>3</SUB>(1 + <I>r</I>)<SUP>3</SUP>] </BLOCKQUOTE><P>The NPV is a third-degree polynomial. The wage can be considered a parameter. Figure 1, above, graphs this polynomial for three specific values of this parameters. In decreasing order, wages are 11/250, 11/500, and 2/250 bushels per person-years for these graphs. </P><P>Given the wage, the IRR is the intersection of the appropriate polynomial with the positive real axis in Figure 1. These IRRs are approximately 101.1%, 175.5%, and 329.5%, respectively. Suppose the economy were competitive, in the sense that capitalists can freely invest and disinvest in any industry. No barriers to entry exist. Then, if this technology is actually in use in producing corn and the wage were the independent variable, the rate of profits would tend to the IRR found for the wage. Profits and losses other than those earned at this rate of profits would be competed away. </P><P>The above graph suggests that, perhaps, the NPV for all wages intersects in two points, one of which is a local maximum. I do not know if this is so. Nor have I thought about why this might be. I guess it is fairly obvious that the local maximum is always at the same rate of profits. The wage drops out of the equation formed by setting the derivative of the NPV, with respect to the rate of profits, to zero. </P><P>I want to focus on the number of crossings of the real axis in the above graph. Figure 2 shows all roots of the polynomial equation defining the NPV. For a maximum wage, the IRR is zero, and it is greater to the right, along the real axis, for a smaller wage. The corresponding real roots, for the maximum wage, are the greatest and least negative rate of profits along the two loci shown in the left half of Figure 2. For smaller wages, these two real roots lie closer together, until around the middle wage used in constructing Figure 1, only one negative, repeated root exists. For any lower wage, the two roots that are not the IRR are complex conjugates. When the wage approaches zero, the workers live on air and all three roots go to (positive or negative) infinity. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><a href="https://1.bp.blogspot.com/-v81qA6QD5xY/WEa-jzFvoyI/AAAAAAAAArc/qVwpc5ueldAtXJzGCOfh715gpzdDZNq2ACLcB/s1600/ThreeRoots.jpg" imageanchor="1" ><img border="0" src="https://1.bp.blogspot.com/-v81qA6QD5xY/WEa-jzFvoyI/AAAAAAAAArc/qVwpc5ueldAtXJzGCOfh715gpzdDZNq2ACLcB/s320/ThreeRoots.jpg" width="320" height="240" /></a></td></tr><tr><td align="center"><b>Figure 2: Multiple Rates of Profit for The Technique</b></td></tr></tbody></table><P>This post has presented an example for thinking about multiple interest rate analysis. It is mainly a matter of raising questions. I do not know how the mathematics for investigating these questions impacts practical applications of multiple interest rate analysis. </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com2tag:blogger.com,1999:blog-26706564.post-28077677505916824002016-11-17T08:19:00.000-05:002016-12-06T08:31:56.598-05:00The Choice Of Technique With Multiple And Complex Interest Rates<P>I have expanded this <A HREF="http://robertvienneau.blogspot.com/2016/10/multiple-and-complex-internal-rates-of.html">post</A> into a <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2869058">working paper</A>. The abstract is: </P><BLOCKQUOTE><B>Abstract:</B> This paper clarifies the relationships between Internal Rates of Return, Net Present Value, and the analysis of the choice of technique in models of production analyzed during the Cambridge capital controversy. Multiple and possibly complex roots of polynomial equations defining the IRR are considered. An algorithm, using these multiple roots to calculate the NPV, justifies the traditional analysis of a reswitching example. </BLOCKQUOTE><P>Michael Osborne, I hope, should find the working paper more constructive than my post. </P><P>(I do not know why, when I delete comments or mark them as spam, they still remain in the upper right.) </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-47773972807246190982016-11-05T11:53:00.000-04:002016-11-07T08:32:12.670-05:00Teaching Calculus To Kids These Days?<B>1.0 Introduction</B><P>A couple of years ago I saw somebody in my local library who was obviously tutoring students in mathematics. I cannot recall how or why, but I started a question. He assured me that advanced high school seniors were taught calculus here. But the approach they teach nowadays does not require kids to learn epsilon-delta definitions of limits and continuity. This surprised me. I understand limits are difficult to wrap one's mind around. For one thing, one needs to not think in terms of dynamics, in some sense. And epsilonic definitions are rarely seen as natural to the beginning student. </P><P>I have since had similar conversations with a few youngsters. And they did not recall epsilon-delta definitions either. I realize that teaching and student recollection varies. Furthermore, the use of epsilon to represent a small distance in the space of the range of the function is a notational convention. Perhaps, some other symbol was used in their classes (although I doubt it). Furthermore, to engineers and practical-oriented students, they might be more interested in getting to problems with derivatives and integrals. (When I asked C. how his calculus class was, he said, "We're still on limits", which I thought expressed an impatience.) </P><P>I wonder about this. I have a theory how some might have justified a change to teaching in calculus since my day, although I can imagine other justifications that do not contradict my ideas below. Anyways, I only intend to raise questions in this post. </P><B>2.0 A Potted History of Calculus after Newton</B><P>When Newton and Liebniz invented the differential calculus, they had a problem with certain quotients. The slope of secants, drawn for two points on a "smooth" function, might be a well-defined ratio. But what does it mean to take a limit? Sometimes Newton seems to treat a denominator as simultaneously zero and non-zero. And this problem with <I>infinitesimals</I> (or fluxions) is compounded when one starts thinking about second derivatives and even higher orders. </P><P>Berkeley quickly pointed out these difficulties. I gather he was concerned to argue against the deism - to him, atheism - that often seemed to accompany Newtonian physics and cosmology. Why criticize the mote in your neighbor's eye without first casting out the beam in your own? Anyways, mathematicians recognized Berkeley had a point about calculus. But the mathematics worked in practice and seemed to be extraordinary useful for physics. </P><P>So mathematicians struggled for centuries, building an immense structure on what they recognized to be an unsound foundation. They also tried to rebuild the foundations. Cauchy, for example, made some improvements. As far as real numbers and limits are concerned, the decisive work came in the second half of the nineteenth century, with Weierstrass' epsilon-delta definitions and Dedekind's construction of the reals out of sets of rational numbers, known as cuts. Whether this was the answer, or whether this just moved the problems deeper down to questions about <A HREF="http://robertvienneau.blogspot.com/2010/08/infinities-of-infinities.html">sets</A> and logic, was not immediately clear. The work of Cantor, Frege, and Russell are of some importance here. The twentieth century saw intensive exploration of such foundational questions. Anyways, nobody seems to have ever found a contradiction in Zermelo-Fraenkel set theory, even if the absence of such contradictions cannot be proven. ZF set theory, with the axiom of choice in many applications, seems to provide a sufficient foundation for the working mathematician. </P><P>I guess that that is how the picture stood around, say, 1960. Newton's own approach to calculus was non-rigorous, but epsilon-delta definitions provide all the rigor introductory students of calculus need. Also, Alfred Tarski had invented something called <A HREF="http://robertvienneau.blogspot.com/2009/04/truth-said-pilate-what-does-that-mean.html">model theory</A>. Along came Abraham Robinson, who used model theory to develop non-standard analysis. Somehow, nonstandard analysis provides a rigorous justification of infinitesimals. (I wouldn't mind understanding the <A HREF="http://mathworld.wolfram.com/Loewenheim-SkolemTheorem.html">L&ouml;wenheim-Skolem theorem</A> either.) </P><P>So maybe it does make sense to teach calculus, without the rigor of epsilon-delta definitions. Keisler wrote a textbook to illustrate the teaching of calculus on the foundations of infinitesimals, maybe easier for the student to understand and justified by the rigor of the advanced abstractions of non-standard analysis. Has this approach, revolutionizing centuries of understanding, won out in introductory calculus classes? </P><B>3.0 Other Special Cases in Introductory Teaching</B><P>I can think of a couple of other cases where what was in my textbooks in calculus and analysis was superseded, in some sense, in more advanced mathematics. I gather mathematical analysis is often informally defined as what the differential and integral calculus would be if taught rigorously. And Rudin (1976) is a standard introduction to analysis. </P><P>Rudin provides an epsilon-delta definition of limits. This definition is more general than you might see in (old?) calculus courses. In such less abstract courses, you might see two definition of limits. One would be for sequences, that is, for functions mapping the natural numbers into the reals. And another would be for functions mapping the real numbers into the real numbers. But Rudin's definition is for functions mapping an arbitrary metric space into (possibly another) arbitrary metric space. One might get the impression that some notion of distance between points is needed to define a limit. But, as was pointed out in the class I took with Rudin as the textbook, a limit of a function is a topological notion. </P><P>A common intuition for integration is as of the area under a curve. This notion can be formalized with the Riemann integral, and, for me, this is the first definition I learned. But another definition, Lebesque integration, is taught in classes on measure theory. Lebesque integrals are more general. Some <A HREF="http://robertvienneau.blogspot.com/2015/03/on-mainstream-economists-ignorance-of.html">functions</A> have a Lebesque integral, but not a Riemann integral. But, if a function has a Riemann integral, it has the same value for the Lebesque integral. </P><P>I offer a suggestion in the spirit of a devil's advocate. Why teach the special case at all in these instances? Why not start with the more general case? Do those who concern themselves with the pedagogy of mathematics selectively advocate the teaching of the more abstract, general case? Is so, how do they choose when this is appropriate? </P><B>4.0 Conclusion</B><P>Is it now quite common - maybe, in the United States - to teach introductory calculus without providing an epsilon-delta definition of a limit? If so, does common justification of this practice draw on a non-standard analysis approach to calculus? Why should this extremely abstract idea influence introductory teaching, but not other abstractions? </P><B>Appendix: Two Definitions of a Limit of a Function and a Theorem</B><P>These are from memory, since I do not want to bother looking them up. The proof of the theorem, probably stated more rigorously, was a test question in a course I took decades ago. </P><BLOCKQUOTE><B>Definition (Metric Space):</B> Let <I>f</I> be a function mapping a metric space <I>X</I> into a metric space <I>Y</I>. <I>L</I> is a limit of <I>f</I> as <I>x</I> approaches <I>x</I><SUB>0</SUB> if and only if, for all &epsilon; &gt; 0, there exists a &delta; &gt; 0 such that, whenever the distance between <I>x</I> and <I>x</I><SUB>0</SUB> is less than &delta;, the distance between <I>f</I>(<I>x</I>) and <I>L</I> is less than &epsilon;. </BLOCKQUOTE><BLOCKQUOTE><B>Definition (Topological):</B> Let <I>f</I> be a function mapping a topological space <I>X</I> into a topological space <I>Y</I>. <I>L</I> is a limit of <I>f</I> as <I>x</I> approaches <I>x</I><SUB>0</SUB> if and only if for all open sets <I>B</I> in <I>Y</I> containing <I>L</I>, the preimage of <I>B</I>, <I>f</I><SUP>-1</SUP>(<I>B</I>), contains <I>x</I><SUB>0</SUB>. </BLOCKQUOTE><BLOCKQUOTE><B>Theorem:</B> Let <I>f</I> map a metric space <I>X</I> into a metric space <I>Y</I>. Then <I>L</I> is a limit of <I>f</I> as <I>x</I> approaches <I>x</I><SUB>0</SUB>, in the metric space definition, if and only if <I>L</I> is also the limit of <I>f</I>, in the topological space definition, in the topologies for <I>X</I> and <I>Y</I> induced by the respective metrics for these spaces. </BLOCKQUOTE><B>References</B><UL><LI>George Berkeley. (1734). <I>The Analyst: A Discourse Address to an Infidel Mathematician...</I> [I never finished this.]</LI><LI>H. Jerome Keisler (1976). <I>Foundations of Infinitesimal Calculus</I>, Prindle, Weber & Schmidt. [I barely started this.]</LI><LI>Morris Kline (1980). <I>Mathematics: The Loss of Certainty</I>, Oxford University Press.</LI><LI>Walter Rudin (1976). <I>Principles of Mathematical Analysis</I>, 3rd edition, McGraw-Hill.</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-86594463317790282502016-10-22T11:51:00.000-04:002016-12-06T08:31:35.573-05:00Multiple And Complex Internal Rates Of Return<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><a href="https://2.bp.blogspot.com/-84R-mUcyNwA/WA3gkWfMiyI/AAAAAAAAAq8/-bCuklbrijgws0bSx9nCMieJuBI2vwKjgCLcB/s1600/AlphaProfitRatesComplexPlane.jpg" imageanchor="1" ><img border="0" src="https://2.bp.blogspot.com/-84R-mUcyNwA/WA3gkWfMiyI/AAAAAAAAAq8/-bCuklbrijgws0bSx9nCMieJuBI2vwKjgCLcB/s320/AlphaProfitRatesComplexPlane.jpg" width="320" height="180" /></a></td></tr><tr><td align="center"><b>Figure 1: One Real and Two Complex Rates of Profit for Alpha Technique</b></td></tr></tbody></table><B>1.0 Introduction</B><P>My intent, in this post, is to refute a few lines in Osborne and Davidson (2016). I want to do this in the spirit of this article, while not denying any valid mathematics. Osborne and Davidson have this to say about the numeric example in Samuelson (1968)<SUP>1</SUP>: </P><BLOCKQUOTE>In other words, when [the Internal Rate of Return] shifts, affecting the capital cost, the product of the unorthodox rates (the duration of the adjusted labor inputs) also shifts such that the overall interest-rate-cost-relationship is linear. This linearity implies that, in the context of this model at least, switching between techniques can happen but reswitching cannot because two straight lines cross only once. Moreover, the relationship between capital cost and the composite interest rate is positive, implying that the neoclassical 'simple tale' that lower rates promote more roundabout technology, is valid when the interest rate is broadly defined. </BLOCKQUOTE><P>Samuelson's example is well-established, and it is incorrect to draw the above conclusion from the Osborne and Davidson model. They derive an equation which, when no pure economic profits exist, relates the price of a consumer good to its cost when a certain composite rate of profits is applied to dated labor inputs. This equation is a tautology; the capital cost on the Right-Hand Side of this equation cannot take on different values without the price on the Left-Hand Side simultaneously varying. Thus, however intriguing this equation may be, it cannot support Osborne and Davidson's supposed refutation of reswitching. </P><B>2.0 A Model</B><P>Consider a flow-input, point-output model of production of, for example, corn. For a given technique of production, let <I>L<SUB>i</SUB></I>, <I>i</I> = 1, ..., <I>n</I>; be the input of labor, measured in person-years, hired <I>i</I> years before the output is produced, for every bushel corn produced. Suppose, for now, that a bushel corn is the numeraire<SUP>2</SUP>. Let the wage, <I>w</I>, be given (in units of bushels per person-year), and suppose wages are advanced. Define: </P><BLOCKQUOTE><I>R</I> = 1 + <I>r</I>, </BLOCKQUOTE><P>where <I>r</I> is the rate of profits. The cost per bushel produced is: </P><BLOCKQUOTE><I>w</I> <I>L</I><SUB>1</SUB> <I>R</I> + <I>w</I> <I>L</I><SUB>2</SUB> <I>R</I><SUP>2</SUP> + ... + <I>w</I> <I>L</I><SUB><I>n</I></SUB> <I>R</I><SUP><I>n</I></SUP></BLOCKQUOTE><P>Define <I>g</I>(<I>R</I>) as the additive inverse of economic profits per bushel produced: </P><BLOCKQUOTE><I>g</I>(<I>R</I>) = <I>w</I> <I>L</I><SUB>1</SUB> <I>R</I> + <I>w</I> <I>L</I><SUB>2</SUB> <I>R</I><SUP>2</SUP> + ... + <I>w</I> <I>L</I><SUB><I>n</I></SUB> <I>R</I><SUP><I>n</I></SUP> - 1 </BLOCKQUOTE><P>Divide through by <I>w</I> <I>L</I><SUB><I>n</I></SUB> to obtain a <I>n</I>th degree polynomial, <I>f</I>(<I>r</I>), with a leading coefficient of unity: </P><BLOCKQUOTE><I>f</I>(<I>R</I>) = <I>R</I><SUP><I>n</I></SUP> + (<I>L</I><SUB><I>n - 1</I></SUB>/<I>L</I><SUB><I>n</I></SUB>) <I>R</I><SUP><I>n - 1</I></SUP> + ... + (<I>L</I><SUB>1</SUB>/<I>L</I><SUB><I>n</I></SUB>) <I>R</I> - 1/(<I>w</I> <I>L</I><SUB><I>n</I></SUB>) </BLOCKQUOTE><P>The Internal Rate of Return (IRR), when this technique is adopted for producing corn, is a zero of this polynomial. </P><B>3.0 A Composite Rate of Profits</B><P>A <I>n</I>th degree polynomial has, in general, <I>n</I> zeros. These zeros need not be positive, non-repeating, or even real. For a polynomial with real coefficients, as above, some of the zeros can be complex conjugate pairs. The IRR is the rate of profits, <I>r</I><SUB>1</SUB>, corresponding to the smallest real zero, <I>R</I><SUB>1</SUB>, exceeding or equal to unity. </P><BLOCKQUOTE><I>r</I><SUB>1</SUB> = <I>R</I><SUB>1</SUB> - 1 &ge; 0 </BLOCKQUOTE><P>The IRR is well-defined only if the wage does not exceed the maximum wage, where the maximum wage is the reciprocal of the sum of dated labor inputs for a bushel corn: </P><BLOCKQUOTE><I>w</I><SUB>max</SUB> = 1/(<I>L</I><SUB>1</SUB> + <I>L</I><SUB>2</SUB> + ... + <I>L</I><SUB><I>n</I></SUB>) </BLOCKQUOTE><P>Let <I>r</I><SUB>2</SUB>, <I>r</I><SUB>3</SUB>, ..., <I>r</I><SUB><I>n</I></SUB> be the other <I>n</I> - 1 zeros of the above polynomial. As I understand it, these zeros, especially any complex ones, are ignored in financial analysis. Notice that these rates of profits are calculated, given the quantities of dated labor inputs and the wage. One cannot consider different rates of profits without varying the wage or vice versa. </P><P>For any complex number <I>z</I>, one can calculate a corresponding real number, namely, the magnitude (or absolute value): </P><BLOCKQUOTE>|<I>z</I>| = |<I>z</I><SUB>real</SUB> + <I>j</I> <I>z</I><SUB>imag</SUB>| = [(<I>z</I><SUB>real</SUB>)<SUP>2</SUP> + (<I>z</I><SUB>imag</SUB>)<SUP>2</SUP>]<SUP>1/2</SUP></BLOCKQUOTE><P>where <I>j</I> is the square root of negative one. (I have been hanging around electrical engineers, who use this notation all the time.) Consider the magnitude of the product of all rates of profits associated with the zeros of the polynomial <I>f</I>(<I>R</I>): </P><BLOCKQUOTE>| <I>r</I><SUB>1</SUB> <I>r</I><SUB>2</SUB> ... <I>r</I><SUB><I>n</I></SUB>| = <I>r</I><SUB>1</SUB> |<I>r</I><SUB>2</SUB>| ... |<I>r</I><SUB><I>n</I></SUB>| </BLOCKQUOTE><P>One can think of this magnitude as a certain composite rate of profits. Michael Osborne's research project, as I understand it, is to explore the meaning and use of this composite rate of profits in a wide variety of models. </P><B>4.0 A Derivation</B><P>One can express any polynomial in terms of its zeros. For <I>f</I>(<I>R</I>), one obtains: </P><BLOCKQUOTE><I>f</I>(<I>R</I>) = (<I>R</I> - <I>R</I><SUB>1</SUB>)(<I>R</I> - <I>R</I><SUB>2</SUB>)...(<I>R</I> - <I>R</I><SUB><I>n</I></SUB>) </BLOCKQUOTE><P>Or: </P><BLOCKQUOTE><I>f</I>(<I>R</I>) = (<I>r</I> - <I>r</I><SUB>1</SUB>)(<I>r</I> - <I>r</I><SUB>2</SUB>)...(<I>r</I> - <I>r</I><SUB><I>n</I></SUB>) </BLOCKQUOTE><P>Two equivalent expressions of the polynomial of interest can be equated: </P><BLOCKQUOTE><I>R</I><SUP><I>n</I></SUP> + (<I>L</I><SUB><I>n - 1</I></SUB>/<I>L</I><SUB><I>n</I></SUB>) <I>R</I><SUP><I>n - 1</I></SUP> + ... + (<I>L</I><SUB>1</SUB>/<I>L</I><SUB><I>n</I></SUB>) <I>R</I> - 1/(<I>w</I> <I>L</I><SUB><I>n</I></SUB>)<BR>= (<I>r</I> - <I>r</I><SUB>1</SUB>)(<I>r</I> - <I>r</I><SUB>2</SUB>)...(<I>r</I> - <I>r</I><SUB><I>n</I></SUB>) </BLOCKQUOTE><P>The above equation holds for any rate of profits. In particular, it holds for a rate of profits equal to zero. Thus, one obtains the following identity: </P><BLOCKQUOTE>1 + (<I>L</I><SUB><I>n - 1</I></SUB>/<I>L</I><SUB><I>n</I></SUB>) + ... + (<I>L</I><SUB>1</SUB>/<I>L</I><SUB><I>n</I></SUB>) - 1/(<I>w</I> <I>L</I><SUB><I>n</I></SUB>) = (-<I>r</I><SUB>1</SUB>)(-<I>r</I><SUB>2</SUB>)...(-<I>r</I><SUB><I>n</I></SUB>) </BLOCKQUOTE><P>Some algebraic manipulation yields: </P><BLOCKQUOTE>(1/<I>w</I>) = (<I>L</I><SUB>1</SUB> + <I>L</I><SUB>2</SUB> + ... + <I>L</I><SUB><I>n</I></SUB>) - <I>L</I><SUB><I>n</I></SUB>(-<I>r</I><SUB>1</SUB>)(-<I>r</I><SUB>2</SUB>)...(-<I>r</I><SUB><I>n</I></SUB>) </BLOCKQUOTE><P>Take the magnitude of both sides. One gets: </P><BLOCKQUOTE>(1/<I>w</I>) = (<I>L</I><SUB>1</SUB> + <I>L</I><SUB>2</SUB> + ... + <I>L</I><SUB><I>n</I></SUB>) + <I>L</I><SUB><I>n</I></SUB><I>r</I><SUB>1</SUB> |<I>r</I><SUB>2</SUB>| ... |<I>r</I><SUB><I>n</I></SUB>| </BLOCKQUOTE><P>The above equation, albeit interesting, is a tautology, expressing the absence of pure economic profits. For a given technique (that is, set of dated labor inputs), one cannot consider independent levels of the two sides of the equation. Osborne and Davidson's mistake is to fail to notice that the tautological nature of the above equation invalidates their use of this equation to say something about the (re)switching of techniques. <P>The Left Hand Side of the above equation is the cost price of a unit output, in terms of person-years. The Right Hand Side is the sum of two terms. The first is the labor embodied in the production of a commodity. The second term is the first labor input, from the most distant time in the past, costed up at the composite rate of profits. Somehow or other, that composite rate of profits, as Osborne and Davidson note, expresses something about the number of time periods over which that first input of labor is accumulated and the distribution of dated labor inputs over those time periods. The number of time periods is expressed in the number of rates of profit that go into forming the composite rate of profits. I find how the distribution of labor inputs affects the composite rate of profits more obscure<SUP>3</SUP>. I also wonder how the composite rate of profits appears for a technique in which a first labor input cannot be found. </P><B>5.0 Numerical Example</B><P>An example might help clarify. Suppose labor inputs, per bushel corn produced, are as in Table 1. </P><TABLE BORDER="1" ALIGN="center"><CAPTION><B>Table 1: The Technology</B></CAPTION><TR><TD ALIGN="center" ROWSPAN="2"><B>Year<BR>Before<BR>Output</B></TD><TD ALIGN="center" COLSPAN="2"><B>Labor Hired for Each Technique</B></TD></TR><TR><TD ALIGN="center"><B>Alpha</B></TD><TD ALIGN="center"><B>Beta</B></TD></TR><TR><TR><TD ALIGN="center"><B>1</B></TD><TD ALIGN="center">33 Person-Years</TD><TD ALIGN="center">0 Person-Years</TD></TR><TR><TD ALIGN="center"><B>2</B></TD><TD ALIGN="center">0 Person-Years</TD><TD ALIGN="center">52 Person-Years</TD></TR><TR><TD ALIGN="center"><B>3</B></TD><TD ALIGN="center">20 Person-Years</TD><TD ALIGN="center">0 Person-Years</TD></TR></TABLE><P></P><B>5.1 Alpha Technique</B><P>The number of time periods, <I>n</I>, for the alpha technique, is three. The polynomial whose zeros are sought is: </P><BLOCKQUOTE><I>f</I><SUB>&alpha;</SUB>(<I>R</I>) = <I>R</I><SUP>3</SUP> + (33/20)<I>R</I> - 1/(20 <I>w</I>) </BLOCKQUOTE><P>The maximum wage is (1/53) bushels per person-years. The above polynomial, not having a term for <I>R</I><SUP>2</SUP>, is a particularly simple form of a cubic equation. Nevertheless, I choose not to write explicit algebraic expressions for its zeros. Instead, consider the complex plane, as graphed in Figure 1, above. The traditional rate of profits is on the half of the real axis extending to the right from zero. The other two zeros are on the rays shown extending to the northwest and southwest. When the wage is at its maximum, the traditional rate of profits is zero and the complex rates of profits are at the rightmost points on those rays, as close as they ever come to zero. For wages below the maximum and above zero, the rates of profits are correspondingly further away from the origin. Figure 2, on the other hand, graphs the traditional and composite rates of profits, as functions of the wage. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><a href="https://4.bp.blogspot.com/-UkmWe6t4Xgo/WA3gd1iPgPI/AAAAAAAAAq4/oK1aOr4CmSk8-jNRavYd01UFILBl-yPUACLcB/s1600/AlphaProfitRates.jpg" imageanchor="1" ><img border="0" src="https://4.bp.blogspot.com/-UkmWe6t4Xgo/WA3gd1iPgPI/AAAAAAAAAq4/oK1aOr4CmSk8-jNRavYd01UFILBl-yPUACLcB/s320/AlphaProfitRates.jpg" width="320" height="180" /></a></td></tr><tr><td align="center"><b>Figure 2: Rate of Profits and Composite Rate of Profits for Alpha Technique</b></td></tr></tbody></table><P></P><B>5.2 Beta Technique</B><P>For the beta technique, the number of time periods, <I>n</I>, is two. The polynomial whose zeros are sought is: </P><BLOCKQUOTE><I>f</I><SUB>&beta;</SUB>(<I>R</I>) = <I>R</I><SUP>2</SUP> - 1/(52 <I>w</I>) </BLOCKQUOTE><P>For wages not exceeding 1/52 bushels per person-year, the traditional rate of profits is: </P><BLOCKQUOTE><I>r</I><SUB>1, &beta;</SUB> = 1/(52 <I>w</I>)<SUP>1/2</SUP> - 1 </BLOCKQUOTE><P>The other rate of profits is: </P><BLOCKQUOTE><I>r</I><SUB>2, &beta;</SUB> = -1/(52 <I>w</I>)<SUP>1/2</SUP> - 1 </BLOCKQUOTE><P>The composite rate of profits is: </P><BLOCKQUOTE><I>r</I><SUB>1, &beta;</SUB> | <I>r</I><SUB>2, &beta;</SUB> | = [1/(52 <I>w</I>)] - 1 </BLOCKQUOTE><P>The dependence of the composite rate of profits on the wage is clearly visible in the beta technique. </P><B>5.3 Cost Minimization</B><P>Figure 3 graphs the traditional and composite rate of profits, as a function of the wage. In the traditional analysis, the cost-minimizing technique is found by choosing the technique on the outer envelope for the two curves to the left in the figure. Although I do not what meaning to assign to it, one could also form the outer envelope for the two curves on the right, that is, the composite rate of profits. If the (composite) rate of profits is zero, the technique on the outer envelope is the one that intersects the wage axis furthest to the right. This is the technique with the smallest total of dated labor inputs, that is, the beta technique. The outer envelope for both the traditional and composite rate of profits yield the same conclusion. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><a href="https://1.bp.blogspot.com/-cJ3rbZ7wZ-U/WA3gXbTwTpI/AAAAAAAAAq0/-C4U8dlYIjI0wBebLuPW2XF1A8AR2da0QCLcB/s1600/FactorPriceFrontier.jpg" imageanchor="1" ><img border="0" src="https://1.bp.blogspot.com/-cJ3rbZ7wZ-U/WA3gXbTwTpI/AAAAAAAAAq0/-C4U8dlYIjI0wBebLuPW2XF1A8AR2da0QCLcB/s320/FactorPriceFrontier.jpg" width="320" height="180" /></a></td></tr><tr><td align="center"><b>Figure 3: Wage-Rate of Profits Curves</b></td></tr></tbody></table><P>If one based the choice of technique on the composite rate of profits, one would find the alpha technique preferable for all composite rate of profits above a small rate. This would be a switching example, not a reswitching example. There would only be one switch point, as shown on the diagram. And, by the traditional analysis, it is indeed a reswitching example, with switch points at <I>r</I><SUB>1</SUB> equal to 10% and 50%. I still see no reason to believe otherwise or to accept a non-equivalent model. </P><B>6.0 Conclusion</B><P>Although I reject Osborne and Davidson's conclusion about reswitching, I find the concept of the composite rate of profits intriguing. I suspect Osborne is more interested in impacting corporate finance, with the Cambridge Capital Controversy being a by-the-way kind of application. I do not see how the composite rate of profit helps with the analysis of the choice of technique. Osborne (2010) uses the composite rate of profits to clarify the relationship between the Internal Rate of Return and Net Present Value. I like that in my previous <A HREF="http://robertvienneau.blogspot.com/2009/02/another-reswitching-example.html">exposition</A> of the above example, I applied an algorithm in which both IRRs and NPVs are relevant. I have not yet absorbed Osborne's NPV analysis. </P><B>Footnotes</B><OL><LI>I have an <A HREF="http://robertvienneau.blogspot.com/2009/02/another-reswitching-example.html">example</A> with reswitching at more reasonable rates of profits.</LI><LI>Osborne and Davidson take a person-year of labor as the numeraire. I do not see anything in this model can depend on which commodity is the numeraire.</LI><LI>Osborne and Davidson state that the composite rate of profits describes the weighted-average timing of labor inputs. Unlike this average, the Austrian average period of production was originally meant to be defined without references to prices.</LI></OL><B>Bibliography</B><UL><LI>Micheal Osborne (2010). A resolution to the NPV-IRR debate? <I>Quarterly Review of Economics and Finance</I>, V. 50, Iss. 2 (May): pp. 234-239 (<A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=522722">working paper</A>).</LI><LI>Michael Osborne (2014). <I>Multiple Interest Rate Analysis: Theory and Applications</I>, Palgrave Macmillan [I HAVE NOT READ THIS].</LI><LI>Michael Osborne and Ian Davidson (2016). The Cambridge capital controversies: contributions from the complex plane, <I>Review of Political Economy</I>, V. 28, No. 2: pp. 251-269.</LI><LI>Paul Samuelson (1968). A summing up, <I>Quarterly Journal of Economics</I>, V. 80, No. 4: pp. 568-583.</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-23302977624817184692016-10-08T12:16:00.000-04:002016-10-08T12:18:42.300-04:00Why Republicans in the USA are "The stupid party"<B>1.0 Introduction</B><P>In 1865, <A HREF="http://robertvienneau.blogspot.com/2008/04/j-s-mill-on-method.html">John</A> <A HREF="http://robertvienneau.blogspot.com/2011/03/some-british-nineteenth-century.html">Stuart</A> <A HREF="http://robertvienneau.blogspot.com/2008/03/against-supply-and-demand.html">Mill</A>, when he was almost 60, was elected to Parliament. He represented the radical wing of the Liberal party. He had been a public intellectual for decades, with lots of books, editorials, and articles for the Tories to draw on in attacking him. Some Tories overreached. This led to the conservative party becoming known as "The stupid party". </P><B>2.0 Adventures in Parliament</B><P>I find Mill's attitude towards being a Member of Parliament (MP) unusual, albeit consistent with his stated opinions. He was not interested in giving speeches in support of his party's view when many others were willing to do so. He "in general reserved [him]self for work which no others were likely to do." (from his <I>Autobiography</I>. Uncited quotes below are from this book.) He had such opportunities, for few radicals were in Parliament. (Earlier in his life, such a group was known in Britain as the Philosophical Radicals.) </P><P>Despite his radicalism, some of his advocacy was in opposition "to what then was, and probably still is, regarded as the advanced liberal opinion". For example, Mill was against abolishing capital punishment and "in favour of seizing enemies' goods in neutral vessels". </P><P>But other efforts seem more progressive, when viewed from the standpoint of later times. In a speech on Gladstone's Reform Bill, Mill argued for sufferage of the working class. He also promoted women's sufferage through his parliamentary work. He put out a pamphlet for reforming British rule in Ireland, including "for settling the land question by giving to existing tenants a permanent tenure, at a fixed rent." He joined in an organization that attempted to have British officers in Jamaica prosecuted, in a criminal case. These officers had engaged in killing, flogging, and general brutality, under the pretence of having civilians brought before court-martials. </P><B>3.0 <I>Considerations on Representative Government</I></B><P>J. S. Mill had long been what we would call a public intellectual. I want to particularly focus on his book with the above title. He gives a qualitative discussion of particular <A HREF="http://robertvienneau.blogspot.com/2016/04/math-is-power.html">voting</A> <A HREF="http://robertvienneau.blogspot.com/2016/06/getting-greater-weight-for-your-vote.html">games</A>. Mill was for proportional representation, also known then as "personal representation". And Mill recommended <A HREF="https://en.wikipedia.org/wiki/Thomas_Hare_%28political_scientist%29">Thomas Hare</A> on the topic. Other issues he considered include: </P><UL><LI>Provide multiple votes (a greater weight) to more highly educated members of the electorate.</LI><LI>Giving voters multiple votes for distributing in elections for a district that had multiple members to elect to a council.</LI><LI>Working class and women's sufferage.</LI><LI>The advantages and disadvantages of a secret ballot (as opposed to an open one).</LI><LI>The advantages and disadvantages of having a two-stage election (e.g., the electoral college, Senators being elected by a state's legislature.</LI><LI>The advantages and disadvantages of an upper house (e.g., the Senate, the House of Lords), under various assumptions about its composition.</LI><LI>Whether or not the chief executive should be independently elected (e.g., the President of the United States) or by the legislature (e.g., the Prime Minister in the United Kingdom).</LI><LI>How the central government and localities should interact and what should the authority and responsibility of each be.</LI></UL><P>In short, Mill seems to write about concerns often of interest today in analytical political science, albeit in a qualitative way and grounded in concrete practices in his time. </P><B>4.0 Attention and the Aftermath</B><P>The Tories in Parliament took advantage of Mill's long paper trail. In debates, they would ask if he wanted to defend some of his previous written statements. Because of Mill's forthrightness, this strategy backfired: </P><BLOCKQUOTE>"My position in the House was further improved... by an ironical reply to some Tory leaders who had quoted against me certain passages of my writings, and called me to account for others, especially for one in 'Considerations on Representative Government,' which said that the Conservative party was, by the law of its composition, the stupidest party. They gained nothing by drawing attention to the passage, which up to that time had not excited any notice, but the <I>sobriquet</I> of 'the stupid party' stuck to them for a considerable time afterwards." </BLOCKQUOTE><P><I>Considerations on Representative Government</I> contains this passage: </P><BLOCKQUOTE><P>"...It is an essential part of democracy that minorities should be adequately represented. No real democracy, nothing but a false show of democracy, is possible without it. </P><P>Those who have seen and felt, in some degree, the force of these considerations, have proposed various expedients by which the evil may be, in greater or lesser degree, mitigated. Lord John Russell, in one of his Reform Bills, introduced a provision that certain constituencies should return three members, and that in these each elector should be allowed to vote only for two; and Mr. Disraeli, in the recent debates, revived the memory of the fact by reproaching him for it, being of opinion, apparently, that it befits a Conservative statesman to regard only means, and to disown scornfully all fellow-feeling with any one who is betrayed, even once, into thinking of ends." </P></BLOCKQUOTE><P>And that passage has this footnote (which I read as noting the existence of negative partisanship): </P><BLOCKQUOTE>"his blunder of Mr. Disraeli (from which, greatly to his credit, Sir John Pakington took an opportunity soon after of separating himself) is a speaking instance, among many, how little the Conservative leaders understand Conservative principles. Without presuming to require from political parties such an amount of virtue and discernment as they that they should comprehend, and know when to apply, the principles of their opponents, we may yet say that it would be a great improvement if each party understood and acted upon its own. Well would it be for England if Conservatives voted consistently for every thing conservative, and Liberals for every thing liberal. We should not then have to wait long for things which, like the present and many other great measures, are eminently both the one and the other. The Conservatives, as <I>being by the law of their existence the stupidest part</I>, have much the greatest sins of this description to answer for; and it is a melancholy truth, that if any measure were proposed on any subject truly, largely, and far-sightedly conservative, even if Liberals were willing to vote for it, the great bulk of the Conservative party would rush blindly in and present it from being carried." (emphasis added.) </BLOCKQUOTE><P>I assume Mill's refers to the following statement, in parliamentary debates, as his "ironical reply": </P><BLOCKQUOTE>"I did not mean that Conservatives are generally stupid; I meant, that stupid persons are generally Conservative. I believe that to be so obvious and undeniable a fact that I hardly think any honourable Gentleman will question it." </BLOCKQUOTE><B>5.0 Conclusion</B><P>And so, to this day, the more conservative party in some countries, such as the United States, is sometimes called "The stupid party". </P><B>References</B><UL><LI>J. S. Mill (1861). <I>Considerations on Representative Government</I></LI><LI>J. S. Mill (1873). <I>Autobiography of John Stuart Mill</I></LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-57221390465555735592016-09-24T12:16:00.000-04:002016-11-07T08:35:25.113-05:00Parliamentary Parties In A Presidential System and the Failure of the Principle of Subsidiarity<B>1.0 Introduction</B><P>Some have argued that the Republican Party, in the United States of American, has been acting, since Newt Gringrich's speakership of the house, more like a parliamentary party<SUP>1</SUP>. And that this creates tensions in a presidential system<SUP>2</SUP>, like the USA. I think I have located another tension that, so far as I know, had not been previously identified when I started this post, months ago<SUP>3</SUP>. </P><P>People line up in local elections often for local reasons, to pursue local interests. In mass publics, even the political leaders in town, district, city, county, and municipal systems cannot be expected be knowledgeable about national issues and political ideology. In big-tent parties, the aggregate of such local movements need not form coherent ideologies. But when at least one national party is dominated by ideological beliefs, local politics might tend to be seen through an ideological lens. Not only might local political bickering become more bitter and rancorous, local politicians might become less responsive to specific characteristics of their areas. Federalism will work worse. Delegating decisions to the lowest authority possible among municipalities, states, and nations does not necessarily lead to more democratically responsive decisions, in some sense. </P><B>2.0 Local Politics</B><P>County-level splits in big-tent political parties do not result in ideological shifts. Suppose there are both right and left wings in two dominant political parties in a country, and these ideological spectra overlap. One party might be more dominant in one region than another. How urban and rural populations; ethnic groups; landholders, financiers, industrialists, professionals, small business owners, and workers line up might vary among regions. Once, say, in the 1950s, the Democrats were the party in the USA of southern whites and urban ethnic immigrants from southeastern Europe. And Republicans were simultaneously the party of African-Americans and big business<SUP>4</SUP>. Supporting a party at a local level, switching sides, and so on need not reflect strong ideological view in such circumstances. It could be a matter of simply seeking more resources for an interest group. </P><P>Once upon a time in Chicago, the Democratic party was extremely dominant, and the party was run like many another big city machine. Harold Washington was a successful reform candidate who became major. The old-time machine politicians had to go somewhere, and they became Republicans. Whatever local tensions were involved in it, this kind of local party split and reforming of one party need not align with any national movement. </P><P>The county I live in has two urban centers. As I understand it, the Democrats are traditionally dominant in the larger city, and the Republicans are dominant in mine. We have had in both cities, in my memory, mayors that were either independent - in the sense, that they ran on neither party line - or bipartisan, in that he ran on both. </P><P>So there are two examples of alignments in local politics that might be said to be more about interest groups, and less about ideological movements. Politics in the USA has been becoming more ideological and falling along a one-dimensional continuum (Hare & Poole, 2013). And I think that has affected local politics. </P><B>Update:</B><P>Rogers (2016) does show that state legislative elections are dominated by national politics. But he does not show any break in such trends with national politics becoming more partisan. But I have stumbled upon Abramowitz and Webster (2015), which support my thesis. I do not know of any literature investigating national effects on local elections, as I postulate. </P><B>Footnotes</B><OL><LI>For this post, I am more interested in the first paragraph in the following quotation. The second paragraph is probably the most widely quoted passage from this book, partly because of Ornstein's standing among right-leaning think tanks and partly because of an accompanying <I>Washington Post</I> editorial:</LI><BLOCKQUOTE><P>"...we identify two overriding sources of dysfunction. The first is the serious mismatch between the political parties, which have become as vehemently adversarial as parliamentary parties, and a governing system that, unlike a parliamentary democracy, makes it extremely difficult for majorities to act. Parliamentary-style parties in a separation-of-powers government are a formula for willful obstruction and policy irresolution. Sixty years ago, Austin Ranney, an eminent political scientist, wrote a prophetic dissent to a famous report by an American Political Science Association committee entitled 'Toward a More Responsible Two-Party System.' The report, by prominent political scientists frustrated with the role of conservative Southern Democrats in blocking civil rights and other social policy, issued a clarion call for more ideologically coherent, internally unified, and adversarial parties in the fashion of a Westminister-style parliamentary democracy like Britain or Canada. Ranney powerfully argued that such parties would be a disaster within the American constitutional system, given our separation of powers, separately elected institutions, and constraints on majority rule that favor cross-party coalitions and compromise. Time has proven Ranney dead right - we now have the kinds of parties the report desired, and it is disastrous. </P><P>The second is the fact that, however awkward it may be for the traditional press and nonpartisan analysts to acknowledge, one of the two major parties, the Republican Party, has become an insurgent outlier - ideologically extreme; contemptuous of the inherited social and economic policy regime; scornful of compromise; unpersuaded by conventional understanding of facts, evidence, and science; and dismissive of the legitimacy of its political opposition. When one party moves this far from the center of American politics, it is extremely difficult to enact policies responsive to the country's most pressing challenges." -- Thomas Mann and Norman Ornstein (2012). </P></BLOCKQUOTE><LI>From an agenda-setting paper on the differences between presidential and parliamentary systems:</LI><BLOCKQUOTE><P>"...the president's strong claim to democratic, even plebiscitarian, legitimacy [stands out]... Following ...Walter Bagehot, ... a presidential system endows the incumbent with both the 'ceremonial' functions of a head of state and the 'effective' functions of a chief executive, thus creating an aura, a self-image, and a set of popular expectations which are all quite different from those associated with a prime minister, no matter how popular he may be. </P><P>But what is most striking is that in a presidential system, the legislators, especially when they represent cohesive, disciplined parties that offer clear ideological and political alternatives, can also claim democratic legitimacy... [W]hen a majority of the legislature represents a political option opposed to the... president...[,] who has the stronger claim to speak on behalf of the people: the president or the legislative majority that opposes his policies? ... One might argue that the United States has successfully rendered such conflicts 'normal' and thus defused them... [T]he uniquely diffuse character of American political parties - which ironically, exasperates many American political scientists and leads them to call for responsible, ideologically disciplined parties - has something to do with it... [T]he development of modern political parties, particular in socially and ideologically polarized countries, generally exacerbates, rather than moderates, conflicts between the legislative and the executive." -- Juan Linz (1990): pp. 53-54.' </P></BLOCKQUOTE><LI>But see Steven Rogers' <A HREF="http://m.ann.sagepub.com/content/667/1/207.full.pdf">study</A>, highlighted by a Jeff Stein <A HREF="http://www.vox.com/2016/9/5/12712932/american-state-government-federalism">article</A> at <I>Vox</I>.</LI><LI>These are tendencies. It is part of my point that such tendencies might be violated, at some time in some specific locality.</LI></OL><B>Selected References</B><UL><LI>Alan Abrsmowitz and Steven Webster (2015). <A HREF="http://www.stevenwwebster.com/research/all_politics_is_national.pdf">All politics is national: The rise of negative partisanship and nationalization of U.S. House and Senate elections in the 21st century.</A></LI><LI>Christopher Hare and Keith T. Poole (2013). <I>The Polarization of Contemporary American Politics</I>.</LI><LI>Matt Grossmann and David A. Hopkins (2015). Ideological Republicans and Group Interest Democrats: The Asymmetry of American Politics, <I>Perspectives on Politics</I>, V. 13, No. 1 (Mar.): pp. 119-139.</LI><LI>Juan J. Linz (1990). The Perils of Presidentialism, <I>Journal of Democracy</I>, V. 1, No. 1 (Winter): pp. 51-69.</LI><LI>Thomas E. Mann and Norman J. Ornstein (2012). <I>It's Even Worse than It Looks: How the American Constitutional System Collided with the New Politics of Extremism</I>, Basic Books.</LI><LI>Steven Rogers (2016). National Forces in State Legislative Elections, <I>AAPSS</I> (Sep.): pp. 207-225.</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-30648401372203190462016-09-09T07:53:00.000-04:002016-09-12T06:21:43.463-04:00Tim Lewins: "Economics, Intelligent-Design Theory, And Homeopathy"<P>Tim Lewens has written a popular introduction to the philosophy of science, <I>The Meaning of Science: An Introduction to the Philosophy of Science.</I> In his first substantial chapter, he writes about what distinguishes science from non-science. Karl Popper and the demarcation problem arise here. He needs examples of near sciences: </P><BLOCKQUOTE>Consider the trio of economics, intelligent-design theory, and homeopathy. The only thing that unites these three endeavors is that their scientific status is regularly questioned in ways that provoke stormy debate. Is economics a science? On the one hand, like many sciences, it oozes both mathematics and authority. On the other hand it is poor at making predictions, and many of its practitioners are surprisingly blase&eacute; when it comes to finding out about how real people think and behave. They would rather build models that tell us what would happen, under simplified circumstances, if people were perfectly rational. So perhaps economics is less like science, and more akin to <I>The Lord of the Rings</I> with equations: it is a mathematically sophisticated exploration of an invented world not much like our own. </BLOCKQUOTE><P>In a later chapter, Lewens recognize that economics is a diverse discipline. He writes about some interesting analyses in economics. And then we get: </P><BLOCKQUOTE>In contrast to these empirically rich forms of economic inquiry [associated with Sen and Kahneman], much work in neoclassical economics is concerned with the largely theoretical analysis of how markets would work if they were populated with individuals endowed with perfect rationality - in other words, creatures of fantasy. We might be tempted to classify these areas of economics as science fiction. Alternatively, we might think that this brand of economics tells us not how the world is but how the world ought to be, if only people would think straight... </BLOCKQUOTE><P>I think Lewens is more complimentary to homeopathy than he is to economics. (He does have a bit more to say about economics than I have quoted.) Controlled experiments in medicine, I gather, consider one intervention as applied to a population. Advocates of homeopathic medicine claim to be treating a whole, particular person in a way which cannot be easily analyzed such reductionist experiments. This, no matter how hostile you may be to it, is an interesting claim for a philosopher to consider. Maybe what they advocate are placebos. Suppose you have a patient that is skeptical of big medicine. Would he react better to a placebo if it is administered in an alternative setting? What, ethically, could such a practitioner say when prescribing extremely diluted "medicine"? </P><P>I still am of the <A HREF="http://robertvienneau.blogspot.com/2013/02/against-science-reality-and-free-will.html">opinion</A> that labelling a claim in economics as "science" or "non-science" should neither add nor subtract to its plausibility, over and above whatever empirical evidence and disciplinary arguments already do. </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-69794360547652933052016-07-29T08:03:00.000-04:002016-07-29T08:03:00.449-04:00Emmanuelle Benicourt Influenced By Steve Keen?<P>I am thinking of absurdity number 3 below. I go a little further because I am amused by the well-established point with which I end this quotation. </P><BLOCKQUOTE><P>"<B>ABSURDITY N<SUP>o</SUP>3 'For a price-taking firm, the demand curve for its own output is a horizontal line at the market price'</B> (Unit 8.3) </P><P>This is <B>false</B>: the demand curve of a price-taking firm <I>is not</I>, and cannot be, horizontal: a firm supply, even if it is 'tiny', affects the price and then the demand of the good it produces.</P><P>The correct assumption should be that the firm <I>believes</I> that the demand curve is horizontal - an erroneous belief, but that is another story... </P><P>In their seminal article, Existence of an Equilibrium for a Competitive Economy, Kenneth Arrow and G&eacute;rard Debreu don't mention agents' beliefs but they, </P><BLOCKQUOTE>'...<I>instruct each production and consumption unit to behave as if</I> the announcement of price <I>p</I> were the equilibrium value' (point 1.4.1, [Benicourt's] italics)</BLOCKQUOTE><P><B>ABSURDITY N<SUP>o</SUP>4 All agents are price-takers (competitive equilibrium)</B></P><P>...Now, any reasonable person will immediately ask: if <I>all</I> agents are price-takers, who set[s] prices? The <I>e-Book</I> answers (implicitly) this question with a circular reasoning... </P><P><I>Conclusion</I>: 'A competitive market', as defined in the <I>CORE e-Book</I>, is not 'an approximation' of any existing market. It is not: </P><BLOCKQUOTE>'...hard to find evidence of perfect competition' (Unit 8.3). </BLOCKQUOTE><P><I>It is impossible</I>. </P><P>The so-called 'competitive economy' model doesn't 'describe an idealised market structure' (Unit 8, p 44). It is not 'unrealistic' - any model is, by definition - <I>it is irrelevant</I>. In fact, it has <I>nothing to do</I> with capitalism. It can be considered, at most, as a variant of market-socialism models, with a benevolent planner setting prices, adding supplies and demands, etc." -- Emmanuelle Benicourt (2016). Is the <I>CORE e-Book</I> a possible solution to our problems? <I>Real-World Economics Review</I>, iss. no. 75, p. 135-142.</P></BLOCKQUOTE>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-10824139869247723732016-06-28T12:24:00.000-04:002016-06-28T12:24:24.684-04:00Getting Greater Weight For Your Vote May Not Give You Relatively More Power<B>1.0 Introduction</B><P>This post presents a perhaps surprising example of results from <A HREF="http://robertvienneau.blogspot.com/2016/04/math-is-power.html">measuring political power</A> in a system with weighted voting. I provide examples in which the weight of a person's vote is increased. Yet that voter, in some cases, gains no additional power, in some sense. In one case, by the measures of voting power considered here, the additional weight has no effect on the power of any voter. In another case, another player, with unchanged weight to his vote, is elevated in power with the voter whose weight is increased. </P><P>I find these results to be an interesting consequence of power measures. I have not yet found a simple example where the effect on the ranking of voting power is different for the three indices considered here. Nor have I found an example where a voter declines in power with an increase in the weight of his vote. </P><B>2.0 An Example of a Voting Game</B><P>A voting game is specified as a set of players, the number of votes needed to enact a bill into law (also referred to as passing a proposition), and the weights for the votes of each player. In considering voting games with a small number of players and weighted, unequal votes, one might think of such a game as describing a council or board of directors, where members represent blocs or geographic districts of varying sizes. </P><P>As example, consider a set, <I>P</I>, of four players, indexed from 0 through 3: </P><BLOCKQUOTE><I>P</I> = The set of players = {0, 1, 2, 3} </BLOCKQUOTE><P>A common way to indicate the remaining parameters for a voting game is a tuple in which the first element is followed by a colon and the remaining elements are separated by commas: </P><BLOCKQUOTE>(6: 4, 3, 2, 1) </BLOCKQUOTE><P>The positive integer before the colon indicates the number of votes - six, in this case - needed to pass a proposition. The remaining integers are the weights of players' votes. In this case, the weight of Player 0's vote is 4, the weight of Player 1's vote is 3, and so on. </P><B>3.0 Two Power Indices</B><P>Consider all 16 possible subsets of the four players. These subsets are listed in the first column of Table 1. A subset of players is labeled a coalition. The second column indicates whether or not the coalition for that row has enough weighted votes to pass a proposition. If so, the characteristic function for that coalition is assigned the value unity. Otherwise, it gets the value zero. A player is decisive for a coalition if the player leaving the coalition will convert it from a winning to a losing coalition. The last four columns in Table 1 have entries of unity for each player that is decisive for each coalition. The last row in Table 1 provides a count, for each player, of the number of coalitions in which that player is decisive. The Penrose-Banzhaf power index, for each player, is the ratio of this total to the number of coalitions. </P><table align="CENTER" border=""><tbody><CAPTION><b>Table 1: Calculations for Penrose-Banzhaf Power Index</b></CAPTION><TR align="CENTER"><TD ROWSPAN="2"><B>Coalition</B></TD><TD ROWSPAN="2"><B>Characteristic<BR>Function</B></TD><TD COLSPAN="4"><B>Player</B></TD></TR><TR align="CENTER"><TD><B>0</B></TD><TD><B>1</B></TD><TD><B>2</B></TD><TD><B>3</B></TD></TR><TR align="CENTER"><TD>{}</TD><TD><I>v</I>( {} ) = 0</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>{0}</TD><TD><I>v</I>( {0} ) = 0</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>{1}</TD><TD><I>v</I>( {1} ) = 0</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>{2}</TD><TD><I>v</I>( {2} ) = 0</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>{3}</TD><TD><I>v</I>( {3} ) = 0</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>{0, 1}</TD><TD><I>v</I>( {0, 1} ) = 1</TD><TD>1</TD><TD>1</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>{0, 2}</TD><TD><I>v</I>( {0, 2} ) = 1</TD><TD>1</TD><TD>0</TD><TD>1</TD><TD>0</TD></TR><TR align="CENTER"><TD>{0, 3}</TD><TD><I>v</I>( {0, 3} ) = 0</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>{1, 2}</TD><TD><I>v</I>( {1, 2} ) = 0</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>{1, 3}</TD><TD><I>v</I>( {1, 3} ) = 0</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>{2, 3}</TD><TD><I>v</I>( {2, 3} ) = 0</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>{0, 1, 2}</TD><TD><I>v</I>( {0, 1, 2} ) = 1</TD><TD>1</TD><TD>0</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>{0, 1, 3}</TD><TD><I>v</I>( {0, 1, 3} ) = 1</TD><TD>1</TD><TD>1</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>{0, 2, 3}</TD><TD><I>v</I>( {0, 2, 3} ) = 1</TD><TD>1</TD><TD>0</TD><TD>1</TD><TD>0</TD></TR><TR align="CENTER"><TD>{1, 2, 3}</TD><TD><I>v</I>( {1, 2, 3} ) = 1</TD><TD>0</TD><TD>1</TD><TD>1</TD><TD>1</TD></TR><TR align="CENTER"><TD>{0, 1, 2, 3}</TD><TD><I>v</I>( {0, 1, 2, 3} ) = 1</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD COLSPAN="2"><B>Total:</B></TD><TD>5</TD><TD>3</TD><TD>3</TD><TD>1</TD></TR></tbody></table><P>The Shapley-Shubik power index considers the order in which players enter a coalition. For the example, one considers all 24 permutations for the players. The first column in Table 2 lists these permutation. For each row, a player gets an entry of unity in the appropriate one of the last four columns if including that player in a coalition, reading the entries in a permutation from left to right, creates a winning coalition. The Shapley-Shubik power index, for each player, is the ratio of the totals of each of the last four columns to the number of permutations. </P><table align="CENTER" border=""><tbody><CAPTION><b>Table 2: Calculations for the Shapley-Shubik Power Index</b></CAPTION><TR align="CENTER"><TD ROWSPAN="2"><B>Permutation</B></TD><TD COLSPAN="4"><B>Player</B></TD></TR><TR align="CENTER"><TD><B>0</B></TD><TD><B>1</B></TD><TD><B>2</B></TD><TD><B>3</B></TD></TR><TR align="CENTER"><TD>(0, 1, 2, 3)</TD><TD>0</TD><TD>1</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>(0, 1, 3, 2)</TD><TD>0</TD><TD>1</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>(0, 2, 1, 3)</TD><TD>0</TD><TD>0</TD><TD>1</TD><TD>0</TD></TR><TR align="CENTER"><TD>(0, 2, 3, 1)</TD><TD>0</TD><TD>0</TD><TD>1</TD><TD>0</TD></TR><TR align="CENTER"><TD>(0, 3, 1, 2)</TD><TD>0</TD><TD>1</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>(0, 3, 2, 1)</TD><TD>0</TD><TD>0</TD><TD>1</TD><TD>0</TD></TR><TR align="CENTER"><TD>(1, 0, 2, 3)</TD><TD>1</TD><TD>0</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>(1, 0, 3, 2)</TD><TD>1</TD><TD>0</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>(1, 2, 0, 3)</TD><TD>1</TD><TD>0</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>(1, 2, 3, 0)</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD>1</TD></TR><TR align="CENTER"><TD>(1, 3, 0, 2)</TD><TD>1</TD><TD>0</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>(1, 3, 2, 0)</TD><TD>0</TD><TD>0</TD><TD>1</TD><TD>0</TD></TR><TR align="CENTER"><TD>(2, 0, 1, 3)</TD><TD>1</TD><TD>0</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>(2, 0, 3, 1)</TD><TD>1</TD><TD>0</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>(2, 1, 0, 3)</TD><TD>1</TD><TD>0</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>(2, 1, 3, 0)</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD>1</TD></TR><TR align="CENTER"><TD>(2, 3, 0, 1)</TD><TD>1</TD><TD>0</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>(2, 3, 1, 0)</TD><TD>0</TD><TD>1</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>(3, 0, 1, 2)</TD><TD>0</TD><TD>1</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>(3, 0, 2, 1)</TD><TD>0</TD><TD>0</TD><TD>1</TD><TD>0</TD></TR><TR align="CENTER"><TD>(3, 1, 0, 2)</TD><TD>1</TD><TD>0</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>(3, 1, 2, 0)</TD><TD>0</TD><TD>0</TD><TD>1</TD><TD>0</TD></TR><TR align="CENTER"><TD>(3, 2, 0, 1)</TD><TD>1</TD><TD>0</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>(3, 2, 1, 0)</TD><TD>0</TD><TD>1</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD><B>Total:</B></TD><TD>10</TD><TD>6</TD><TD>6</TD><TD>2</TD></TR></tbody></table><P></P><B>4.0 Three Power Indices for Three Voting Games</B><P>Table 3 summarizes and expands on the above calculations. The Penrose-Banzhaf power index need not sum over the players to unity. Accordingly, I break this index down into two indices, where the second index is normalized. The Shapley-Shubik power index is guaranteed to sum to unity. I introduce two other voting games, with corresponding power indices, presented in Tables 4 and 5. </P><table align="CENTER" border=""><tbody><CAPTION><b>Table 3: Power Indices for (6: 4, 3, 2, 1)</b></CAPTION><TR align="CENTER"><TD ROWSPAN="2"><B>Player</B></TD><TD COLSPAN="2"><B>Penrose-Banzhaf Power Index</B></TD><TD ROWSPAN="2"><B>Shapley-Shubik<BR>Power Index</B></TD></TR><TR align="CENTER"><TD><B>Index</B></TD><TD><B>Normalized</B></TD></TR><TR align="CENTER"><TD>0</TD><TD>5/16</TD><TD>5/12</TD><TD>10/24 = 5/12</TD></TR><TR align="CENTER"><TD>1</TD><TD>3/16</TD><TD>3/12 = 1/4</TD><TD>6/24 = 1/4</TD></TR><TR align="CENTER"><TD>2</TD><TD>3/16</TD><TD>3/12 = 1/4</TD><TD>6/24 = 1/4</TD></TR><TR align="CENTER"><TD>3</TD><TD>1/16</TD><TD>1/12</TD><TD>2/24 = 1/12</TD></TR></tbody></table><P></P><table align="CENTER" border=""><tbody><CAPTION><b>Table 4: Power Indices for (6: 4, 2, 2, 1)</b></CAPTION><TR align="CENTER"><TD ROWSPAN="2"><B>Player</B></TD><TD COLSPAN="2"><B>Penrose-Banzhaf Power Index</B></TD><TD ROWSPAN="2"><B>Shapley-Shubik<BR>Power Index</B></TD></TR><TR align="CENTER"><TD><B>Index</B></TD><TD><B>Normalized</B></TD></TR><TR align="CENTER"><TD>0</TD><TD>6/16 = 3/8</TD><TD>6/10 = 3/5</TD><TD>16/24 = 2/3</TD></TR><TR align="CENTER"><TD>1</TD><TD>2/16 = 1/8</TD><TD>2/10 = 1/5</TD><TD>4/24 = 1/6</TD></TR><TR align="CENTER"><TD>2</TD><TD>2/16 = 1/8</TD><TD>2/10 = 1/5</TD><TD>4/24 = 1/6</TD></TR><TR align="CENTER"><TD>3</TD><TD>0</TD><TD>0</TD><TD>0</TD></TR></tbody></table><P></P><table align="CENTER" border=""><tbody><CAPTION><b>Table 5: Power Indices for (5: 4, 2, 2, 1)</b></CAPTION><TR align="CENTER"><TD ROWSPAN="2"><B>Player</B></TD><TD COLSPAN="2"><B>Penrose-Banzhaf Power Index</B></TD><TD ROWSPAN="2"><B>Shapley-Shubik<BR>Power Index</B></TD></TR><TR align="CENTER"><TD><B>Index</B></TD><TD><B>Normalized</B></TD></TR><TR align="CENTER"><TD>0</TD><TD>6/16 = 3/8</TD><TD>6/12 = 1/2</TD><TD>12/24 = 1/2</TD></TR><TR align="CENTER"><TD>1</TD><TD>2/16 = 1/8</TD><TD>2/12 = 1/6</TD><TD>4/24 = 1/6</TD></TR><TR align="CENTER"><TD>2</TD><TD>2/16 = 1/8</TD><TD>2/12 = 1/6</TD><TD>4/24 = 1/6</TD></TR><TR align="CENTER"><TD>3</TD><TD>2/16 = 1/8</TD><TD>2/12 = 1/6</TD><TD>4/24 = 1/6</TD></TR></tbody></table><P></P><B>5.0 Constitutional Changes</B><P>Consider a change in the constitution, from one of the three voting games with tables in the previous section to another such game. The calculations allow one to measure the impact on voting power for any such change. To simplify matters, I consider only rankings of voting power. And, for these three voting games, the three power indices consider here happen to yield the same ranks, for any given voting game out of these three. </P><P>Accordingly, Table 6 shows changes in the rules (the "constitution") for these cases. The change to the rules on the right superficially strengthens Player 1, either by increasing the weight of Player 1's vote or requiring less votes to pass a resolution. As noted below, I am unsure what naive intuition might be for the second row. For the third vote, the number of votes needed to pass a proposition is altered such that a simple majority is needed before and after the change in weight. </P><table align="CENTER" border=""><tbody><CAPTION><b>Table 6: Changing the Rules to Strengthen the Players?</b></CAPTION><TR align="CENTER"><TD><B>Starting Game</B></TD><TD><B>Player Ranks</B></TD><TD><B>Ending Game</B></TD><TD><B>Player Ranks</B></TR><TR align="CENTER"><TD>(6: 4, 2, 2, 1)</TD><TD ROWSPAN="2">0 &gt; 1 = 2 &gt; 3</TD><TD>(6: 4, 3, 2, 1)</TD><TD>0 &gt; 1 = 2 &gt; 3</TD></TR><TR align="CENTER"><TD>(6: 4, 2, 2, 1)</TD><TD>(5: 4, 2, 2, 1)</TD><TD>0 &gt; 1 = 2 = 3</TD></TR><TR align="CENTER"><TD>(5: 4, 2, 2, 1)</TD><TD>0 &gt; 1 = 2 = 3</TD><TD>(6: 4, 3, 2, 1)</TD><TD>0 &gt; 1 = 2 &gt; 3</TD></TR></tbody></table><P>The first row shows a case where the weight of Player 1's vote increases, which might intuitively give him more power with respect to the apparently weaker Players 2 and 3. Yet this increase in weight also increases the power of Players 2 and 3, even though the weight of their votes does not change. And Player 1 remains equal in power to Player 2, both before and after the change. In fact, the change has no effect on the ranking of the players' voting power. </P><P>The second row shows a case where the votes needed to pass a measure declines, after the change in rules, from a super-majority to a simple majority, given the total of weighted votes. Would one expect such a constitutional amendment to strengthen the most powerful, or moderately powerful voters before the change? I find that this change raises the power of the weakest voter to the power of the middling voters. I am not sure this is counter-intuitive, unlike the other two rows. </P><P>The third row shows a case in which, like the first row, the weight of Player 1's vote increases. Both before and after the change, a simple majority, given the total of weighted votes, is needed to pass a proposition. This change makes Player 1 more powerful than the weakest player, as one might intuitively expect. But Player 2 is also made more powerful than the weakest player, despite the weight of his vote not varying. And Player 1 ends up no more powerful than Player 2. These effects on Player 2 seem counter-intuitive to me. </P><B>6.0 Conclusions</B><P>So my examples above have presented somewhat counter-intuitive results in voting games. </P><P>I gather that the Deegan-Packel and Holler-Packel are some other power indices I might find of interest. And Straffin (1994) is one paper that explains axioms that characterize some power index or other. </P><B>References</B><UL><LI>Donald P. Green and Ian Shapiro (1996). <I>Pathologies of Rational Choice Theory: A Critique of Applications in Political Science</I>, Yale University Press</LI><LI>P. Straffin (1994). Power and stability in politics. <I>Handbook of Game Theory with Economic Applications</I>, V. 2, Elsevier.</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-57695521856736222812016-06-15T12:57:00.000-04:002016-06-16T06:06:27.759-04:00The History and Sociology of Game Theory: A Reading List<P>For me, this list is aspirational. I've read Mirowski and the Weintraub-edited book. I've just checked the Erickson book out of a library. </P><UL><LI>S. M. Amadae (2016). <A HREF="https://www.amazon.com/Prisoners-Reason-Neoliberal-Political-Economy/dp/1107671191"><I>Prisoners of Reason: Game Theory and Neoliberal Political Economy</I></A>, Cambridge University Press.</LI><LI>Paul Erickson (2015). <A HREF="https://www.amazon.com/World-Game-Theorists-Made/dp/022609717X"><I>The World the Game Theorists Made</I></A>, University of Chicago Press.</LI><LI>Robert Leonard (2010). <A HREF="https://www.amazon.com/Neumann-Morgenstern-Creation-Game-Theory/dp/052156266X"><I>Von Neumann, Morgenstern, and the Creation of Game Theory: From Chess to Social Science, 1900-1960</I></A>, Cambridge University Press.</LI><LI>Philip Mirowski (2002). <A HREF="https://www.amazon.com/Machine-Dreams-Economics-Becomes-Science/dp/0521775264"><I>Machine Dreams: Economics Becomes a Cyborg Science</I></A>, Cambridge University Press.</LI><LI>E. Roy Weintraub (editor) (1995). <A HREF="https://www.amazon.com/Toward-History-Political-Economy-Supplement/dp/0822312530"><I>Toward a History of Game Theory</I></A>, Duke University Press.</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com1tag:blogger.com,1999:blog-26706564.post-39214107481571649722016-05-16T20:07:00.000-04:002016-05-16T20:07:50.813-04:00A Turing Machine for a Binary Counter<table align="CENTER" border=""><tbody><CAPTION><b>Table 1: Tape in Successive Start States</b></CAPTION><TR align="CENTER"><TD><B>Input/Output Tape</B></TD><TD><B>Decimal</B></TD></TR><TR align="CENTER"><TD>t<B>b</B>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>t<B>b</B>1</TD><TD>1</TD></TR><TR align="CENTER"><TD>t<B>b</B>10</TD><TD>2</TD></TR><TR align="CENTER"><TD>t<B>b</B>11</TD><TD>3</TD></TR><TR align="CENTER"><TD>t<B>b</B>100</TD><TD>4</TD></TR><TR align="CENTER"><TD>t<B>b</B>101</TD><TD>5</TD></TR></tbody></table><P></P><B>1.0 Introduction</B><P>This post describes another <A REF="http://robertvienneau.blogspot.com/2016/05/a-turing-machine-for-calculating.html">program</A> for a <A REF="http://robertvienneau.blogspot.com/2016/04/a-steam-experience-for-flash-mob.html">Turing Machine</A>. This Turing machine implements a binary counter (Table 1). I do not think I am being original here. (Maybe this was in the textbook on computability and automata that I have been reading.) </P><B>2.0 Alphabet</B><P></P><table align="CENTER" border=""><tbody><CAPTION><b>Table 2: The Alphabet For The Input Tape</b></CAPTION><TR align="CENTER"><TD><B>Symbol</B></TD><TD><B>Number Of<BR>Occurrences</B></TD><TD><B>Comments</B></TD></TR><TR align="CENTER"><TD>t</TD><TD>1</TD><TD>Start-of-tape Symbol</TD></TR><TR align="CENTER"><TD>b</TD><TD>Potentially Infinite</TD><TD>Blank</TD></TR><TR align="CENTER"><TD>0</TD><TD>Potentially Infinite</TD><TD>Binary Digit Zero</TD></TR><TR align="CENTER"><TD>1</TD><TD>Potentially Infinite</TD><TD>Binary Digit One</TD></TR></tbody></table><P></P><B>3.0 Specification of Valid Input Tapes</B><P></P><P>At start, the (input) tape should contain, in this order: </P><UL><LI>t, the start-of-tape symbol.</LI><LI>b, a blank.</LI><LI>A sequence of binary digits, with a length of at least one.</LI></UL><P>The above specification allows for any number of unnecessary leading zeros in the binary number on the tape. The head shall be at the blank following the start-of-tape symbol. </P><B>4.0 Specification of State</B><P>The machine starts in the Start state. Error is the only halting state. Table 3 describes some conditions, for a non-erroneous input tape, that states are designed to satisfy, on entry and exit. For the states GoToEnd, FindZero, CreateTrailingOne, Increment, and ResetHead, the Turing machine may experience many transitions that leaves the machine in that state after the state has been entered. When the state PauseCounter has been entered, the next increment of a binary number appears on the tape. </P><table align="CENTER" border=""><tbody><CAPTION><b>Table 3: States</b></CAPTION><TR align="CENTER"><TD ROWSPAN="2"><B>State</B></TD><TD COLSPAN="2"><B>Selected Conditions</B></TD></TR><TR align="CENTER"><TD><B>On Entry</B></TD><TD><B>On Exit</B></TD></TR><TR align="LEFT"><TD>Start</TD><TD>The head is immediately to the left of the binary number on the tape. (The binary number on the tape at this point is referred to as "the original binary number" below.)</TD><TD>Same as the entry condition for GoToEnd.</TD></TR><TR align="LEFT"><TD>GoToEnd</TD><TD>The head is under the first digit of the binary number on the tape.</TD><TD>Same as the entry condition for FindZero.</TD></TR><TR align="LEFT"><TD>FindZero</TD><TD>The head is under the last digit of the binary number on the tape</TD><TD>If all digits in the original binary number are 1 and that number has not been updated with a leading zero, the head is under the first digit of the binary number on the tape. If the original binary number contained at least one digit 0, the head is under the location of the last instance of 0 in the original binary number, and that digit has been changed to a 1. Otherwise, the head is under the first digit in the binary number now on the tape, and that digit is now a 1 (having once been a leading zero).</TD></TR><TR align="LEFT"><TD>CreateLeadingZero</TD><TD>All the digits in the original binary number are 1. The head is under the first digit of the binary number on the tape.</TD><TD>Same as the entry condition for CreateTrailingOne</TD></TR><TR align="LEFT"><TD>CreateTrailingOne</TD><TD>All the digits in the original binary number are 1. The first digit in the original binary number has been replaced by 0. The head is under that first digit.</TD><TD>The original binary number has been shifted one digit to the left, and a leading zero has been prepended to it. The head is under the last digit of the binary number now on the tape.</TD></TR><TR align="LEFT"><TD>StepForward</TD><TD>If all digits in the original binary number are 1, that number has been shifted one digit to the left, that number has been updated with a leading 0 which is now a 1, and the head is under that digit. Otherwise, the last instance of 0 in the original number has been updated to a 1, and the head is now under that digit tape.</TD><TD>Same as the entry condition for Increment.</TD></TR><TR align="LEFT"><TD>Increment</TD><TD>If all digits in the original binary number are 1, that number has been shifted one digit to the left, that number has been updated with a leading 0 which is now a 1, and the head is under the next location on the tape. Otherwise, the last instance of 0 in the original number has been updated to a 1, and the head is now under the next location on the tape.</TD><TD>Same as the entry condition for ResetHead. All the 1's to the right of the 0 updated to a 1 have themselves been updated to a 0.</TD></TR><TR align="LEFT"><TD>ResetHead</TD><TD>The head is under the last digit of the binary number on the tape, and that number is the successor of the original binary number.</TD><TD>Same as the entry condition for PauseCounter.</TD></TR><TR align="LEFT"><TD>PauseCounter</TD><TD>The head is immediately to the left of the binary number on the tape, and that number is the successor of the original binary number.</TD><TD></TD></TR></tbody></table><P>I think one could express the conditions in the above lengthy table as logical predicates. And one could develop a formal proof that the state transition rules in the appendix ensure that these conditions are met on entry and exit of the non-halting tape, at least for non-erroneous input tapes. I do not quite see how invariants would be used here. (When trying to think rigorously about source code, I attempt to identify invariants for loops.) </P><B>5.0 Length of Tape and the Number of States</B><P>Suppose the state PauseCounter was a halting state. Then this Turing machine would be a linear bounded automaton. In the Chomsky hierarchy, automata that accept context-sensitive languages need not be more general than linear bound automata. </P><P>The program for this Turing machine consists of 10 states. The number of characters on the tape grows at the rate O(log<SUB>2</SUB> <I>n</I>), where <I>n</I> is the number of cycles through the start state. I gather the above instructions could be easily modified to not use any start-of-tape symbol. Anyways, 20 people seems more than sufficient for the <A HREF="http://robertvienneau.blogspot.com/2016/04/a-steam-experience-for-flash-mob.html">group activity</A> I have defined, for this particular Turing machine. </P><B>Appendix A: State Transition Tables</B><P>This appendix provides detail specification of state transition rules for each of the non-halting states. I provide these rules by tables, with each table showing a pair of states. </P><table align="CENTER" border=""><tbody><CAPTION><b>Table A-1: Start and GoToEnd</b></CAPTION><TR align="CENTER"><TD COLSPAN="3"><B>Start</B></TD><TD></TD><TD COLSPAN="3"><B>GoToEnd</B></TD></TR><TR align="CENTER"><TD><B>t</B></TD><TD>t</TD><TD>Error</TD><TD></TD><TD><B>t</B></TD><TD>t</TD><TD>Error</TD></TR><TR align="CENTER"><TD><B>b</B></TD><TD>Forwards</TD><TD>GoToEnd</TD><TD></TD><TD><B>b</B></TD><TD>Backwards</TD><TD>FindZero</TD></TR><TR align="CENTER"><TD><B>0</B></TD><TD>0</TD><TD>Error</TD><TD></TD><TD><B>0</B></TD><TD>Forwards</TD><TD>GoToEnd</TD></TR><TR align="CENTER"><TD><B>1</B></TD><TD>1</TD><TD>Error</TD><TD></TD><TD><B>1</B></TD><TD>Forwards</TD><TD>GoToEnd</TD></TR></tbody></table><table align="CENTER" border=""><tbody><CAPTION><b>Table A-2: FindZero and CreateLeadingZero</b></CAPTION><TR align="CENTER"><TD COLSPAN="3"><B>FindZero</B></TD><TD></TD><TD COLSPAN="3"><B>CreateLeadingZero</B></TD></TR><TR align="CENTER"><TD><B>t</B></TD><TD>t</TD><TD>Error</TD><TD></TD><TD><B>t</B></TD><TD>t</TD><TD>Error</TD></TR><TR align="CENTER"><TD><B>b</B></TD><TD>Forwards</TD><TD>CreateLeadingZero</TD><TD></TD><TD><B>b</B></TD><TD>b</TD><TD>Error</TD></TR><TR align="CENTER"><TD><B>0</B></TD><TD>1</TD><TD>StepForward</TD><TD></TD><TD><B>0</B></TD><TD>0</TD><TD>Error</TD></TR><TR align="CENTER"><TD><B>1</B></TD><TD>Backwards</TD><TD>FindZero</TD><TD></TD><TD><B>1</B></TD><TD>0</TD><TD>CreateTrailingOne</TD></TR></tbody></table><table align="CENTER" border=""><tbody><CAPTION><b>Table A-3: CreateTrailingOne and StepForward</b></CAPTION><TR align="CENTER"><TD COLSPAN="3"><B>CreateTrailingOne</B></TD><TD></TD><TD COLSPAN="3"><B>StepForward</B></TD></TR><TR align="CENTER"><TD><B>t</B></TD><TD>t</TD><TD>Error</TD><TD></TD><TD><B>t</B></TD><TD>t</TD><TD>Error</TD></TR><TR align="CENTER"><TD><B>b</B></TD><TD>1</TD><TD>FindZero</TD><TD></TD><TD><B>b</B></TD><TD>b</TD><TD>Error</TD></TR><TR align="CENTER"><TD><B>0</B></TD><TD>Forwards</TD><TD>CreateTrailingOne</TD><TD></TD><TD><B>0</B></TD><TD>0</TD><TD>Error</TD></TR><TR align="CENTER"><TD><B>1</B></TD><TD>Forwards</TD><TD>CreateTrailingOne</TD><TD></TD><TD><B>1</B></TD><TD>Forwards</TD><TD>Increment</TD></TR></tbody></table><table align="CENTER" border=""><tbody><CAPTION><b>Table A-4: Increment and ResetHead</b></CAPTION><TR align="CENTER"><TD COLSPAN="3"><B>Increment</B></TD><TD></TD><TD COLSPAN="3"><B>ResetHead</B></TD></TR><TR align="CENTER"><TD><B>t</B></TD><TD>t</TD><TD>Error</TD><TD></TD><TD><B>t</B></TD><TD>t</TD><TD>Error</TD></TR><TR align="CENTER"><TD><B>b</B></TD><TD>Backwards</TD><TD>ResetHead</TD><TD></TD><TD><B>b</B></TD><TD>b</TD><TD>PauseCounter</TD></TR><TR align="CENTER"><TD><B>0</B></TD><TD>Forwards</TD><TD>Increment</TD><TD></TD><TD><B>0</B></TD><TD>Backwards</TD><TD>ResetHead</TD></TR><TR align="CENTER"><TD><B>1</B></TD><TD>0</TD><TD>Increment</TD><TD></TD><TD><B>1</B></TD><TD>Backwards</TD><TD>ResetHead</TD></TR></tbody></table><table align="CENTER" border=""><tbody><CAPTION><b>Table A-5: PauseCounter</b></CAPTION><TR align="CENTER"><TD COLSPAN="3"><B>PauseCounter</B></TD></TR><TR align="CENTER"><TD><B>t</B></TD><TD>t</TD><TD>Error</TD></TR><TR align="CENTER"><TD><B>b</B></TD><TD>b</TD><TD>Start</TD></TR><TR align="CENTER"><TD><B>0</B></TD><TD>0</TD><TD>Error</TD></TR><TR align="CENTER"><TD><B>1</B></TD><TD>1</TD><TD>Error</TD></TR></tbody></table>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-86422781669673250682016-05-14T10:31:00.000-04:002016-05-14T10:31:30.734-04:00Choice Of Technique And Search Models Of Labor Markets<P>I do not have an analysis or example to go with this post title. I suggest this would be an interesting research topic. What are implications of the analysis of the choice of technique, if any, for search models of labor markets? </P><P>Consider the neoclassical theory of supply and demand in labor markets under perfect competition. We know (Opocher and Steedman 2015, Vienneau 2005) that that theory is fatally undermined by an analysis of cost-minimizing firms. </P><P>I have recently read an overview, by Steve Fleetwood (2016), of <A HREF="http://robertvienneau.blogspot.com/2010/10/nobel-prize-for-epicycles-in-labor.html">models</A> of search and matching in labor markets. And he illustrates these models with graphs of two crossing monotone curves that, at a glance, look much like labor supply and demand curves. But these curves are drawn in a different space and have a different rationale and derivation than labor supply and demand curves. A wage curve is graphed with the job creation curve. The abscissa is the tightness of the labor market, as measured by the ratio of vacancies to unemployment. The ordinate is the wage, as in the mistaken introductory story. The wage curve is also graphed against the Beveridge curve in a different space, namely, with the present discounted value of expected profit from a vacant job against unemployment. </P><P>In a long run analysis, a higher wage is associated with a lower rate of profits. This wage-rate of profits curves has implications for present discounted values. I do not see why an analysis inspired by Sraffa could not undermine search models. But one would have to go further than this to confirm this intuition. And one would need to read some of the original literature. </P><P>I do not claim that search models might not have some use in a reconstituted economics. </P><B>Reference</B><UL><LI>Steve Fleetwood (2016). Reflections upon neoclassical labour economics, in <I>What is Neoclassical Economics? Debating the origins, meaning and significance</I>, (ed. by Jamie Morgan), Routledge.</LI><LI>Arrigo Opocher and Ian Steedman (2015). <A HREF="http://www.cambridge.org/us/academic/subjects/economics/microeconomics/full-industry-equilibrium-theory-industrial-long-run"><I>Full Industry Equilibrium: A Theory of the Industrial Long Run</I></A>, Cambridge University Press.</LI><LI>Robert L. Vienneau (2005). <A HREF="http://onlinelibrary.wiley.com/doi/10.1111/j.1467-9957.2005.00467.x/abstract">On Labour Demand and Equilibria of the Firm</A>, <I>Manchester School</I>, V. 73, Iss. 5: pp. 612-619.</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-73756105584126819482016-05-11T08:01:00.000-04:002016-05-11T08:01:06.447-04:00A Turing Machine For Calculating The Fibonacci Sequence<table align="CENTER" border=""><tbody><CAPTION><b>Table 1: Representation of the Fibonacci Sequence</b></CAPTION><TR align="CENTER"><TD><B>Input/Output Tape</B></TD><TD><B>Terms in Series</B></TD></TR><TR align="CENTER"><TD>0<B>b</B>1;1;</TD><TD>1, 1</TD></TR><TR align="CENTER"><TD>0b1<B>;</B>1;11;</TD><TD>1, 1, 2</TD></TR><TR align="CENTER"><TD>0b1;1<B>;</B>11;111</TD><TD>1, 1, 2, 3</TD></TR><TR align="CENTER"><TD>0b1;1;11<B>;</B>111;11111;</TD><TD>1, 1, 2, 3, 5</TD></TR><TR align="CENTER"><TD>0b1;1;11;111<B>;</B>11111;11111111;</TD><TD>1, 1, 2, 3, 5, 8</TD></TR></tbody></table><P></P><B>1.0 Introduction</B><P>I thought I would describe the program for a specific <A HREF="http://robertvienneau.blogspot.com/2016/04/a-steam-experience-for-flash-mob.html">Turing machine</A>. This Turing machine computes the Fibonacci sequence in tally arithmetic, as illustrated in Table 1 above. The left-hand column shows the tape for the Turing machine for successive transitions into the Start state. (The location of the head is indicated by the bolded character.) The right-hand column shows a more familiar representation of a Fibonacci sequence. This Turing machine never halts for valid inputs. It can calculate other infinite sequences, such as specific Lucas sequences, for other valid inputs. </P><P>A Turing machine is specified by the alphabet of characters that can appear on the tape, possible valid sequences of characters for the start of the tape, the location of the head at the beginning of a computation, the states and the state transition rules, and the location of the state pointer at beginning of a computation. </P><B>2.0 Alphabet</B><P></P><table align="CENTER" border=""><tbody><CAPTION><b>Table 2: The Alphabet For The Input Tape</b></CAPTION><TR align="CENTER"><TD><B>Symbol</B></TD><TD><B>Number Of<BR>Occurrences</B></TD><TD><B>Comments</B></TD></TR><TR align="CENTER"><TD>0</TD><TD>1</TD><TD>Start of tape marker</TD></TR><TR align="CENTER"><TD>b</TD><TD>Potentially Infinite</TD><TD>Blank</TD></TR><TR align="CENTER"><TD>;</TD><TD>Potentially Infinite</TD><TD>Symbol for number termination</TD></TR><TR align="CENTER"><TD>1</TD><TD>Potentially Infinite</TD><TD>A tally</TD></TR><TR align="CENTER"><TD>x</TD><TD>1</TD><TD>For internal use</TD></TR><TR align="CENTER"><TD>y</TD><TD>1</TD><TD>For internal use</TD></TR><TR align="CENTER"><TD>z</TD><TD>1</TD><TD>For internal use</TD></TR></tbody></table><P></P><B>3.0 Specification of Valid Input Tapes</B><P></P><P>At start, the (input) tape should contain, in this order: </P><UL><LI>0, the start of tape marker.</LI><LI>b, a blank.</LI><LI>Zero or more 1s.</LI><LI>;, a semicolon.</LI><LI>One or more of the following:</LI><UL><LI>Zero or more 1s.</LI><LI>;, a semicolon.</LI></UL></UL><P>The head shall be at a blank or semicolon such that exactly two semicolons exist in the tape to the right of the head. Table 3 provides examples (with the head being at the bolded character). </P><table align="CENTER" border=""><tbody><CAPTION><b>Table 3: Examples of Valid Initial Input</b></CAPTION><TR align="CENTER"><TD>0<B>b</B>;;</TD></TR><TR align="CENTER"><TD>0<B>b</B>1;;</TD></TR><TR align="CENTER"><TD>0<B>b</B>1;1;</TD></TR><TR align="CENTER"><TD>0<B>b</B>11;1;</TD></TR><TR align="CENTER"><TD>0b1;1;11;111<B>;</B>11111;11111111;</TD></TR></tbody></table><P></P><B>4.0 Definition of State</B><P>The states are grouped into two subroutines, CopyPair and Add. Error is the only halting state, to be entered when an invalid input tape is detected. The Turing machine begins the computation with the state pointer pointing to the Start state, in the CopyPair subroutine. Eventually, the Turing machine enters the PauseCopy state. The machine then transitions to the StartAdd state, in the Add subroutine. Another number in the sequence has been successfully appended to the tape when the Turing machine enters the PauseAdd state. </P><P>The Turing machine then transitions into the Start state. The CopyPair and Add subroutines are repeated in pairs forever. </P><B>4.1 CopyPair</B><P>The input tape for the CopyPair subroutine is any valid input tape, as described above. The state pointer starts in the Start tape. Error is the only halting state. The subroutine exits with a transition from the PauseCopy state to the StartAdd state. When the PauseCopy state is entered, the tape shall be in the following configuration: </P><UL><LI>The terminal semicolon in the tape, when the Start state was entered, shall be replaced with a z.</LI><LI>The head shall be at that z.</LI><LI>The tape to the right of the z shall contain a copy of the character string to the right of the head when the Start state was entered.</LI></UL><P>This subroutine can be implemented by the states described in Table 4. The detailed implementation of each state is provided in the appendix. Throughout these states, there are transitions to the Error state triggered by encountering on the tape a character that cannot be there in a valid computation. </P><table align="CENTER" border=""><tbody><CAPTION><b>Table 4: States in the CopyPair Subroutine</b></CAPTION><TR align="CENTER"><TD><B>State</B></TD><TD><B>Description</B></TD></TR><TR align="CENTER"><TD>Start</TD><TD>Moves the head forward one character.</TD></TR><TR align="CENTER"><TD>ReadFirstChar</TD><TD>Replaces first ; or 1 (after position of head when the subroutine was called) with x or y, respectively.</TD></TR><TR align="CENTER"><TD>WriteFirstSemi</TD><TD>Writes a ; at the end of the tape. Transitions to GoToTapeEnd.</TD></TR><TR align="CENTER"><TD>WriteFirstOne</TD><TD>Writes a 1 at the end of the tape. Transitions to GoToTapeEnd.</TD></TR><TR align="CENTER"><TD>GoToTapeEnd</TD><TD>Moves the head backward one character to locate the head at the character that was at the end of the tape when the subroutine was called.</TD></TR><TR align="CENTER"><TD>MarkTapeEnd</TD><TD>Replaces original terminating ; with z.</TD></TR><TR align="CENTER"><TD>NexChar</TD><TD>Replaces the x or y on the tape with ; or 1, respectively.</TD></TR><TR align="CENTER"><TD>StepForward</TD><TD>Moves the head forward one character.</TD></TR><TR align="CENTER"><TD>ReadChar</TD><TD>Replaces the next ; or 1 with x or y, respectively.</TD></TR><TR align="CENTER"><TD>WriteSemi</TD><TD>Writes a ; at the end of the tape. Transitions to NextChar.</TD></TR><TR align="CENTER"><TD>WriteOne</TD><TD>Writes a 1 at the end of the tape. Transitions to NextChar.</TD></TR><TR align="CENTER"><TD>WriteLastSemi</TD><TD>Writes a ; at the end of the tape. Transitions to SetHead.</TD></TR><TR align="CENTER"><TD>SetHead</TD><TD>Moves head to the z on the tape.</TD></TR><TR align="CENTER"><TD>PauseCopy</TD><TD>For noting that last two numbers on the tape, when the subroutine was called, have been copied to the end of the tape.</TD></TR></tbody></table><P></P><B>4.2 Add</B><P>When the PauseAdd state is entered, the tape shall be in the following configuration: </P><UL><LI>The semicolon between the z and the last semicolon, when the StartAdd state is entered, shall be replaced by a 1, if there is at least one 1 between this character and the terminating semicolon.</LI><LI>The semicolon at the end of the tape, when the StartAdd state is entered, shall be erased (replaced by a blank).</LI><LI>The character before the erased semicolon shall be replaced by a semicolon.</LI><LI>The z shall be replaced by a semicolon.</LI><LI>The head shall be at a semicolon such that two semicolons exist to the right of the head.</LI></UL><P></P><table align="CENTER" border=""><tbody><CAPTION><b>Table 5: States in the Add Subroutine</b></CAPTION><TR align="CENTER"><TD><B>State</B></TD><TD><B>Description</B></TD></TR><TR align="CENTER"><TD>StartAdd</TD><TD>Moves the head forward one character.</TD></TR><TR align="CENTER"><TD>FindSemiForDele</TD><TD>Replaces the ; mid-number with 1.</TD></TR><TR align="CENTER"><TD>FindSumEnd</TD><TD>Erases terminating ;.</TD></TR><TR align="CENTER"><TD>EndSum</TD><TD>Writes terminating ; at the tape position one character backwards.</TD></TR><TR align="CENTER"><TD>FindSumStart</TD><TD>Replaces z with ;.</TD></TR><TR align="CENTER"><TD>StepBackward</TD><TD>Moves the head backwards one character.</TD></TR><TR align="CENTER"><TD>ResetHead</TD><TD>Set head to previous ;, before the ; just written.</TD></TR><TR align="CENTER"><TD>PauseAdd</TD><TD>For noting next number in Fibonacci series.</TD></TR></tbody></table><P></P><B>5.0 Length of Tape and the Number of States</B><P>After three run-throughs of this Turing machine, five numbers in the Fibonacci sequence will be calculated. And the tape will contain 19 characters. As shown in Table 6, the number of states is 22. For the <A HREF="http://robertvienneau.blogspot.com/2016/04/a-steam-experience-for-flash-mob.html">group activity</A> I have defined for simulating a Turing machine, 42 people are needed. (One more person is needed, in computing the next number in the sequence, to be erased from the tape than ends up as characters on the tape.) I suppose one could get by with 36 people, if one is willing to some represent two states, one in each subroutine. </P><table align="CENTER" border=""><tbody><CAPTION><b>Table 6: State Count</b></CAPTION><TR align="CENTER"><TD><B>Subroutine</B></TD><TD><B>Number Of<BR>States</B></TD><TD><B>State Names</B></TD></TR><TR align="CENTER"><TD>CopyPair</TD><TD>15</TD><TD>Error, Start, ReadFirstChar,<BR>WriteFirstSemi, WriteFirstOne,<BR>GoToTapeEnd, MarkTapeEnd,<BR>NextChar, StepForward,<BR>ReadChar, WriteSemi,<BR>WriteLastSemi, SetHead,<BR>WriteOne, PauseCopy</TD></TR><TR align="CENTER"><TD>Add</TD><TD>7</TD><TD>StartAdd, FindSemiForDele,<BR>FindSumEnd, EndSum,<BR>FindSumStart, StepBackward,<BR>PauseAdd</TD></TR><TR align="CENTER"><TD><B>Total</B></TD><TD>22</TD><TD></TD></TR></tbody></table><P></P><B>Appendix A: State Transition Tables</B><P></P><B>A.1: The CopyPair Subroutine</B><table align="CENTER" border=""><tbody><CAPTION><b>Table A-1: Start and ReadFirstChar</b></CAPTION><TR align="CENTER"><TD COLSPAN="3"><B>Start</B></TD><TD></TD><TD COLSPAN="3"><B>ReadFirstChar</B></TD></TR><TR align="CENTER"><TD><B>0</B></TD><TD>0</TD><TD>Error</TD><TD></TD><TD><B>0</B></TD><TD>0</TD><TD>Error</TD></TR><TR align="CENTER"><TD><B>b</B></TD><TD>Forwards</TD><TD>ReadFirstChar</TD><TD></TD><TD><B>b</B></TD><TD>b</TD><TD>Error</TD></TR><TR align="CENTER"><TD><B>;</B></TD><TD>Forwards</TD><TD>ReadFirstChar</TD><TD></TD><TD><B>;</B></TD><TD>x</TD><TD>WriteFirstSemi</TD></TR><TR align="CENTER"><TD><B>1</B></TD><TD>1</TD><TD>Error</TD><TD></TD><TD><B>1</B></TD><TD>y</TD><TD>WriteFirstOne</TD></TR><TR align="CENTER"><TD><B>x</B></TD><TD>x</TD><TD>Error</TD><TD></TD><TD><B>x</B></TD><TD>x</TD><TD>Error</TD></TR><TR align="CENTER"><TD><B>y</B></TD><TD>y</TD><TD>Error</TD><TD></TD><TD><B>y</B></TD><TD>y</TD><TD>Error</TD></TR><TR align="CENTER"><TD><B>z</B></TD><TD>z</TD><TD>Error</TD><TD></TD><TD><B>z</B></TD><TD>z</TD><TD>Error</TD></TR></tbody></table><table align="CENTER" border=""><tbody><CAPTION><b>Table A-2: WriteFirstSemi and WriteFirstOne</b></CAPTION><TR align="CENTER"><TD COLSPAN="3"><B>WriteFirstSemi</B></TD><TD></TD><TD COLSPAN="3"><B>WriteFirstOne</B></TD></TR><TR align="CENTER"><TD><B>0</B></TD><TD>0</TD><TD>Error</TD><TD></TD><TD><B>0</B></TD><TD>0</TD><TD>Error</TD></TR><TR align="CENTER"><TD><B>b</B></TD><TD>;</TD><TD>GoToTapeEnd</TD><TD></TD><TD><B>b</B></TD><TD>1</TD><TD>GoToTapeEnd</TD></TR><TR align="CENTER"><TD><B>;</B></TD><TD>Forwards</TD><TD>WriteFirstSemi</TD><TD></TD><TD><B>;</B></TD><TD>Forwards</TD><TD>WriteFirstOne</TD></TR><TR align="CENTER"><TD><B>1</B></TD><TD>Forwards</TD><TD>WriteFirstSemi</TD><TD></TD><TD><B>1</B></TD><TD>Forwards</TD><TD>WriteFirstOne</TD></TR><TR align="CENTER"><TD><B>x</B></TD><TD>Forwards</TD><TD>WriteFirstSemi</TD><TD></TD><TD><B>x</B></TD><TD>Forwards</TD><TD>WriteFirstOne</TD></TR><TR align="CENTER"><TD><B>y</B></TD><TD>Forwards</TD><TD>WriteFirstSemi</TD><TD></TD><TD><B>y</B></TD><TD>Forwards</TD><TD>WriteFirstOne</TD></TR><TR align="CENTER"><TD><B>z</B></TD><TD>z</TD><TD>Error</TD><TD></TD><TD><B>z</B></TD><TD>z</TD><TD>Error</TD></TR></tbody></table><table align="CENTER" border=""><tbody><CAPTION><b>Table A-3: GoToTapeEnd and MarkTapeEnd</b></CAPTION><TR align="CENTER"><TD COLSPAN="3"><B>GoToTapeEnd</B></TD><TD></TD><TD COLSPAN="3"><B>MarkTapeEnd</B></TD></TR><TR align="CENTER"><TD><B>0</B></TD><TD></TD><TD></TD><TD></TD><TD><B>0</B></TD><TD>0</TD><TD>Error</TD></TR><TR align="CENTER"><TD><B>b</B></TD><TD></TD><TD></TD><TD></TD><TD><B>b</B></TD><TD>b</TD><TD>Error</TD></TR><TR align="CENTER"><TD><B>;</B></TD><TD>Backwards</TD><TD>MarkTapeEnd</TD><TD></TD><TD><B>;</B></TD><TD>z</TD><TD>NextChar</TD></TR><TR align="CENTER"><TD><B>1</B></TD><TD>Backwards</TD><TD>MarkTapeEnd</TD><TD></TD><TD><B>1</B></TD><TD>1</TD><TD>Error</TD></TR><TR align="CENTER"><TD><B>x</B></TD><TD></TD><TD></TD><TD></TD><TD><B>x</B></TD><TD>x</TD><TD>Error</TD></TR><TR align="CENTER"><TD><B>y</B></TD><TD></TD><TD></TD><TD></TD><TD><B>y</B></TD><TD>y</TD><TD>Error</TD></TR><TR align="CENTER"><TD><B>z</B></TD><TD></TD><TD></TD><TD></TD><TD><B>z</B></TD><TD>z</TD><TD>Error</TD></TR></tbody></table><table align="CENTER" border=""><tbody><CAPTION><b>Table A-4: NextChar and StepForward</b></CAPTION><TR align="CENTER"><TD COLSPAN="3"><B>NextChar</B></TD><TD></TD><TD COLSPAN="3"><B>StepForward</B></TD></TR><TR align="CENTER"><TD><B>0</B></TD><TD>0</TD><TD>Error</TD><TD></TD><TD><B>0</B></TD><TD></TD><TD></TD></TR><TR align="CENTER"><TD><B>b</B></TD><TD>b</TD><TD>Error</TD><TD></TD><TD><B>b</B></TD><TD></TD><TD></TD></TR><TR align="CENTER"><TD><B>;</B></TD><TD>Backwards</TD><TD>NextChar</TD><TD></TD><TD><B>;</B></TD><TD>Forwards</TD><TD>ReadChar</TD></TR><TR align="CENTER"><TD><B>1</B></TD><TD>Backwards</TD><TD>NextChar</TD><TD></TD><TD><B>1</B></TD><TD>Forwards</TD><TD>ReadChar</TD></TR><TR align="CENTER"><TD><B>x</B></TD><TD>;</TD><TD>StepForward</TD><TD></TD><TD><B>x</B></TD><TD></TD><TD></TD></TR><TR align="CENTER"><TD><B>y</B></TD><TD>1</TD><TD>StepForward</TD><TD></TD><TD><B>y</B></TD><TD></TD><TD></TD></TR><TR align="CENTER"><TD><B>z</B></TD><TD>Backwards</TD><TD>NextChar</TD><TD></TD><TD><B>z</B></TD><TD></TD><TD></TD></TR></tbody></table><table align="CENTER" border=""><tbody><CAPTION><b>Table A-5: ReadChar and WriteSemi</b></CAPTION><TR align="CENTER"><TD COLSPAN="3"><B>ReadChar</B></TD><TD></TD><TD COLSPAN="3"><B>WriteSemi</B></TD></TR><TR align="CENTER"><TD><B>0</B></TD><TD>0</TD><TD>Error</TD><TD></TD><TD><B>0</B></TD><TD>0</TD><TD>Error</TD></TR><TR align="CENTER"><TD><B>b</B></TD><TD>1</TD><TD>Error</TD><TD></TD><TD><B>b</B></TD><TD>;</TD><TD>NextChar</TD></TR><TR align="CENTER"><TD><B>;</B></TD><TD>x</TD><TD>WriteSemi</TD><TD></TD><TD><B>;</B></TD><TD>Fowards</TD><TD>WriteSemi</TD></TR><TR align="CENTER"><TD><B>1</B></TD><TD>y</TD><TD>WriteOne</TD><TD></TD><TD><B>1</B></TD><TD>Forwards</TD><TD>WriteSemi</TD></TR><TR align="CENTER"><TD><B>x</B></TD><TD>x</TD><TD>Error</TD><TD></TD><TD><B>x</B></TD><TD>Forwards</TD><TD>WriteSemi</TD></TR><TR align="CENTER"><TD><B>y</B></TD><TD>y</TD><TD>Error</TD><TD></TD><TD><B>y</B></TD><TD>Forwards</TD><TD>WriteSemi</TD></TR><TR align="CENTER"><TD><B>z</B></TD><TD>z</TD><TD>WriteLastSemi</TD><TD></TD><TD><B>z</B></TD><TD>Forwards</TD><TD>WriteSemi</TD></TR></tbody></table><table align="CENTER" border=""><tbody><CAPTION><b>Table A-6: WriteLastSemi and SetHead</b></CAPTION><TR align="CENTER"><TD COLSPAN="3"><B>WriteLastSemi</B></TD><TD></TD><TD COLSPAN="3"><B>SetHead</B></TD></TR><TR align="CENTER"><TD><B>0</B></TD><TD>0</TD><TD>Error</TD><TD></TD><TD><B>0</B></TD><TD>0</TD><TD>Error</TD></TR><TR align="CENTER"><TD><B>b</B></TD><TD>;</TD><TD>SetHead</TD><TD></TD><TD><B>b</B></TD><TD>b</TD><TD>Error</TD></TR><TR align="CENTER"><TD><B>;</B></TD><TD>Forwards</TD><TD>WriteLastSemi</TD><TD></TD><TD><B>;</B></TD><TD>Backwards</TD><TD>SetHead</TD></TR><TR align="CENTER"><TD><B>1</B></TD><TD>Forwards</TD><TD>WriteLastSemi</TD><TD></TD><TD><B>1</B></TD><TD>Backwards</TD><TD>SetHead</TD></TR><TR align="CENTER"><TD><B>x</B></TD><TD>Forwards</TD><TD>WriteLastSemi</TD><TD></TD><TD><B>x</B></TD><TD>x</TD><TD>Error</TD></TR><TR align="CENTER"><TD><B>y</B></TD><TD>Forwards</TD><TD>WriteLastSemi</TD><TD></TD><TD><B>y</B></TD><TD>y</TD><TD>Error</TD></TR><TR align="CENTER"><TD><B>z</B></TD><TD>Forwards</TD><TD>WriteLastSemi</TD><TD></TD><TD><B>z</B></TD><TD>z</TD><TD>PauseCopy</TD></TR></tbody></table><table align="CENTER" border=""><tbody><CAPTION><b>Table A-7: WriteOne and PauseCopy</b></CAPTION><TR align="CENTER"><TD COLSPAN="3"><B>WriteOne</B></TD><TD></TD><TD COLSPAN="3"><B>PauseCopy</B></TD></TR><TR align="CENTER"><TD><B>0</B></TD><TD>0</TD><TD>Error</TD><TD></TD><TD><B>0</B></TD><TD></TD><TD></TD></TR><TR align="CENTER"><TD><B>b</B></TD><TD>1</TD><TD>NextChar</TD><TD></TD><TD><B>b</B></TD><TD></TD><TD></TD></TR><TR align="CENTER"><TD><B>;</B></TD><TD>Forwards</TD><TD>WriteOne</TD><TD></TD><TD><B>;</B></TD><TD></TD><TD></TD></TR><TR align="CENTER"><TD><B>1</B></TD><TD>Forwards</TD><TD>WriteOne</TD><TD></TD><TD><B>1</B></TD><TD></TD><TD></TD></TR><TR align="CENTER"><TD><B>x</B></TD><TD>Forwards</TD><TD>WriteOne</TD><TD></TD><TD><B>x</B></TD><TD></TD><TD></TD></TR><TR align="CENTER"><TD><B>y</B></TD><TD>Forwards</TD><TD>WriteOne</TD><TD></TD><TD><B>y</B></TD><TD></TD><TD></TD></TR><TR align="CENTER"><TD><B>z</B></TD><TD>Forwards</TD><TD>WriteOne</TD><TD></TD><TD><B>z</B></TD><TD>z</TD><TD>StartAdd</TD></TR></tbody></table><B>A.2: The Add Subroutine</B><table align="CENTER" border=""><tbody><CAPTION><b>Table A-8: StartAdd and FindSemiForDele</b></CAPTION><TR align="CENTER"><TD COLSPAN="3"><B>StartAdd</B></TD><TD></TD><TD COLSPAN="3"><B>FindSemiForDele</B></TD></TR><TR align="CENTER"><TD><B>0</B></TD><TD></TD><TD></TD><TD></TD><TD><B>0</B></TD><TD></TD><TD></TD></TR><TR align="CENTER"><TD><B>b</B></TD><TD></TD><TD></TD><TD></TD><TD><B>b</B></TD><TD></TD><TD></TD></TR><TR align="CENTER"><TD><B>;</B></TD><TD></TD><TD></TD><TD></TD><TD><B>;</B></TD><TD>1</TD><TD>FindSumEnd</TD></TR><TR align="CENTER"><TD><B>1</B></TD><TD></TD><TD></TD><TD></TD><TD><B>1</B></TD><TD>Forwards</TD><TD>FindSemiForDele</TD></TR><TR align="CENTER"><TD><B>x</B></TD><TD></TD><TD></TD><TD></TD><TD><B>x</B></TD><TD></TD><TD></TD></TR><TR align="CENTER"><TD><B>y</B></TD><TD></TD><TD></TD><TD></TD><TD><B>y</B></TD><TD></TD><TD></TD></TR><TR align="CENTER"><TD><B>z</B></TD><TD>Forwards</TD><TD>FindSemiForDele</TD><TD></TD><TD><B>z</B></TD><TD></TD><TD></TD></TR></tbody></table><table align="CENTER" border=""><tbody><CAPTION><b>Table A-9: FindSumEnd and EndSum</b></CAPTION><TR align="CENTER"><TD COLSPAN="3"><B>FindSumEnd</B></TD><TD></TD><TD COLSPAN="3"><B>EndSum</B></TD></TR><TR align="CENTER"><TD><B>0</B></TD><TD></TD><TD></TD><TD></TD><TD><B>0</B></TD><TD></TD><TD></TD></TR><TR align="CENTER"><TD><B>b</B></TD><TD>Backwards</TD><TD>EndSum</TD><TD></TD><TD><B>b</B></TD><TD>Backwards</TD><TD>EndSum</TD></TR><TR align="CENTER"><TD><B>;</B></TD><TD>b</TD><TD>EndSum</TD><TD></TD><TD><B>;</B></TD><TD>b</TD><TD>EndSum</TD></TR><TR align="CENTER"><TD><B>1</B></TD><TD>Forwards</TD><TD>FindSumEnd</TD><TD></TD><TD><B>1</B></TD><TD>;</TD><TD>FindSumStart</TD></TR><TR align="CENTER"><TD><B>x</B></TD><TD></TD><TD></TD><TD></TD><TD><B>x</B></TD><TD></TD><TD></TD></TR><TR align="CENTER"><TD><B>y</B></TD><TD></TD><TD></TD><TD></TD><TD><B>y</B></TD><TD></TD><TD></TD></TR><TR align="CENTER"><TD><B>z</B></TD><TD></TD><TD></TD><TD></TD><TD><B>z</B></TD><TD></TD><TD></TD></TR></tbody></table><table align="CENTER" border=""><tbody><CAPTION><b>Table A-10: FindSumStart and StepBackward</b></CAPTION><TR align="CENTER"><TD COLSPAN="3"><B>FindSumStart</B></TD><TD></TD><TD COLSPAN="3"><B>StepBackward</B></TD></TR><TR align="CENTER"><TD><B>0</B></TD><TD></TD><TD></TD><TD></TD><TD><B>0</B></TD><TD></TD><TD></TD></TR><TR align="CENTER"><TD><B>b</B></TD><TD></TD><TD></TD><TD></TD><TD><B>b</B></TD><TD></TD><TD></TD></TR><TR align="CENTER"><TD><B>;</B></TD><TD>Backwards</TD><TD>FindSumStart</TD><TD></TD><TD><B>;</B></TD><TD>Backwards</TD><TD>ResetHead</TD></TR><TR align="CENTER"><TD><B>1</B></TD><TD>Backwards</TD><TD>FindSumStart</TD><TD></TD><TD><B>1</B></TD><TD></TD><TD></TD></TR><TR align="CENTER"><TD><B>x</B></TD><TD></TD><TD></TD><TD></TD><TD><B>x</B></TD><TD></TD><TD></TD></TR><TR align="CENTER"><TD><B>y</B></TD><TD></TD><TD></TD><TD></TD><TD><B>y</B></TD><TD></TD><TD></TD></TR><TR align="CENTER"><TD><B>z</B></TD><TD>;</TD><TD>StepBackward</TD><TD></TD><TD><B>z</B></TD><TD></TD><TD></TD></TR></tbody></table><table align="CENTER" border=""><tbody><CAPTION><b>Table A-11: ResetHead and PauseAdd</b></CAPTION><TR align="CENTER"><TD COLSPAN="3"><B>ResetHead</B></TD><TD></TD><TD COLSPAN="3"><B>PauseAdd</B></TD></TR><TR align="CENTER"><TD><B>0</B></TD><TD></TD><TD></TD><TD></TD><TD><B>0</B></TD><TD></TD><TD></TD></TR><TR align="CENTER"><TD><B>b</B></TD><TD></TD><TD></TD><TD></TD><TD><B>b</B></TD><TD></TD><TD></TD></TR><TR align="CENTER"><TD><B>;</B></TD><TD>;</TD><TD>PauseAdd</TD><TD></TD><TD><B>;</B></TD><TD>;</TD><TD>Start</TD></TR><TR align="CENTER"><TD><B>1</B></TD><TD>Backwards</TD><TD>ResetHead</TD><TD></TD><TD><B>1</B></TD><TD></TD><TD></TD></TR><TR align="CENTER"><TD><B>x</B></TD><TD></TD><TD></TD><TD></TD><TD><B>x</B></TD><TD></TD><TD></TD></TR><TR align="CENTER"><TD><B>y</B></TD><TD></TD><TD></TD><TD></TD><TD><B>y</B></TD><TD></TD><TD></TD></TR><TR align="CENTER"><TD><B>z</B></TD><TD></TD><TD></TD><TD></TD><TD><B>z</B></TD><TD></TD><TD></TD></TR></tbody></table><B>A.3: Modifications?</B><P>The above is my first working version. I have not proven that cases can never arise where I have not specified rules in the tables for the states for the Add subroutine. Nor do I know that all rules can be triggered by some, possibly invalid, input tape. I know that I have not defined the minimum number of states for the system. For example, the ReadChar state could be defined as in Table A-12, along with the elimination of the WriteLastSemi and SetHead states. This would result in the CopyPair subroutine specification not being met and a tighter coupling between the two subroutines. On the other hand, the subroutines are already coupled through the appearance of z on the tape during the transition from one subroutine to the other. </P><table align="CENTER" border=""><tbody><CAPTION><b>Table A-12: Modified ReadChar</b></CAPTION><TR align="CENTER"><TD COLSPAN="3"><B>ReadChar</B></TD></TR><TR align="CENTER"><TD><B>0</B></TD><TD>0</TD><TD>Error</TD></TR><TR align="CENTER"><TD><B>b</B></TD><TD>1</TD><TD>Error</TD></TR><TR align="CENTER"><TD><B>;</B></TD><TD>x</TD><TD>WriteSemi</TD></TR><TR align="CENTER"><TD><B>1</B></TD><TD>y</TD><TD>WriteOne</TD></TR><TR align="CENTER"><TD><B>x</B></TD><TD>x</TD><TD>Error</TD></TR><TR align="CENTER"><TD><B>y</B></TD><TD>y</TD><TD>Error</TD></TR><TR align="CENTER"><TD><B>z</B></TD><TD>z</TD><TD>PauseCopy</TD></TR></tbody></table>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-37737817891428931102016-05-07T10:57:00.001-04:002016-05-07T10:57:41.796-04:00Noam Chomsky And Norman Mailer Share A Jail Cell For A Night<P>No joke. This happened as a result of an October 1967 march on the Pentagon to protest the Vietnam war. I find I had misremembered this passage. I recalled Mailer as being much less modest, as not acknowledging that technical linguistics used mathematical methods that might be beyond him at that stage of his life, no matter how much time he put into it. (I haven't actually read all of the technical works by Chomsky in the references below.) I have always liked Mailer's reporting and essays better than his novels, an opinion that I probably share with many and that he did not appreciate. </P><BLOCKQUOTE><P>"Definitive word came through. The lawyers were gone, the Commissioners were gone: nobody out until morning. So Mailer picked his bunk. It was next to Noam Chomsky, a slim-featured man with an ascetic expression, and an air of gentle but absolute moral integrity. Friends at Wellfleet had wanted him to meet Chomsky at a summer before - he had been told that Chomsky, although barely thirty, was considered a genius at MIT for his new contributions to linguistics - but Mailer had arrived at the party too late. Now, as he bunked down next to Chomsky, Mailer looked for some way to open a discussion on linguistics - he had an amateur's interest in the subject, no, rather he had a mad inventor's interest, with several wild theories in his pocket which he had never been able to exercise since he could not understand what he read in linguistics books. So he cleared his throat now once or twice, turned over in bed, looked for a preparatory question, and recognized that he and Chomsky might share a cell for months, and be the best and most civilized of cellmates, before the mood would be proper to strike the first note of inquiry into what was obviously the tightly packed conceptual coils of Chomsky's intellections. Instead they chatted mildly of the day, of the arrests (Chomsky had also been arrested with Dellinger), and of when they would get out. Chomsky - by all odds a dedicated teacher - seemed uneasy at the thought of missing class on Monday.</P><P>On that long unwinding passage from the contractions of the day into the deliberations of the dream, Mailer passed through a revery over much traveled and by now level ground where he thought once more of the war in Vietnam, the charges against it, the defenses for it, and his own final condemnation which had landed him here on this filthy blanket and lumpy bed, this smoke-filled barracks air, where he listened half-asleep to the echoes of Teague's loud confident Leninist voice, he, Mailer, ex-revolutionary, now last of the small entrepreneurs, Left Conservative, that lonely flag - there was no one in America who had a position even remotely like his own, who else could indeed could offer such a solution as he possessed to such a war, such a damnable war. Let us leave him as he passes into sleep. The argument in his brain can be submitted to the reader in the following pages with somewhat more order than Mailer possessed on his long voyage out into the unfamiliar dimensions of prison rest..." -- Norman Mailer (1968). </P></BLOCKQUOTE><B>References</B><UL><LI>Noam Chomsky (1959). On certain formal properties of grammars, <I>Information and Control</I>, V. 2: pp. 137-167.</LI><LI>Noam Chomsky (1965). <I>Aspects of the Theory of Syntax</I>, MIT Press.</LI><LI>Noam Chomsky (1969). <I>American Power and the New Mandarins</I>, Pantheon Books.</LI><LI>Noam Chomsky and M. P. Sch&uuml;tzenberger (1963). The algebraic theory of context-free languages, in <I>Computer Programming and Formal Systems</I>, North Holland.</LI><LI>Norman Mailer (1968) <I>The Armies of the Night: History as a Novel, the Novel as History</I>, New American Library.</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com3tag:blogger.com,1999:blog-26706564.post-31083396772773635712016-04-30T11:20:00.000-04:002016-05-07T10:40:29.492-04:00A STEAM Experience For A Flash Mob<B>1.0 Introduction</B><P>STEAM stands for Science, Technology, Engineering, Arts, and Mathematics. This post describes a possible plan for a crowd of many people to participate in. Roles for players consist of: </P><UL><LI>A Recorder.</LI><LI>State Actors.</LI><LI>Holders of letters in a line.</LI></UL><P>I once read Terry Eagleton suggesting that part of the definition of art is that it be "<A HREF="http://www.moma.org/collection/artists/3528">somewhat pointless</A>." </P><B>2.0 Equipment</B><P>Equipment to be provided consists of: </P><UL><LI>A six-sided die.</LI><LI>Two balls. They could be soccer balls, beach balls, volley balls, or so on. One ball is called the Head, and the other ball is called the State Pointer.</LI><LI>Six sets of equipment, labelled 1 through 6. A set of equipment consists of:</LI><UL><LI>A set of cards, where each card is a "letter" from an alphabet. Letters can be, for example, "Blank", "(", ")", ";", "End", "0", and "1". Many letters must have many cards with that letter.</LI><LI>A set of state placards. Each state placard contains:</LI><UL><LI>An arbitrary label. These labels are arbitrary, but not repeated. They could be in high elvish, for all it matters, as long as participants can pronounce each label.</LI><LI>Either the word "Halt" or a set of rules. The placards for the halting states may also contain a short phrase. Each rule in a set of rules is designated by a letter from the alphabet.</LI></UL><LI>Guidelines for setting up. These guidelines include:</LI><UL><LI>Optional guidelines for the geographical distribution of states.</LI><LI>A specification of which State Actor initially holds the State Pointer.</LI><LI>Guidelines for forming an initial line of letters from the alphabet. These guidelines must include a specification of which holder of a letter initially also holds the Head.</LI></UL></UL></UL><B>3.0 Playing the Game</B><P></P><B>3.1 Preliminaries</B><P>The Recorder throws the die and chooses the corresponding set of equipment. One might create only one set of equipment, and this step would be omitted. </P><P>The Recorder distributes the state placards. A audience member comes up for each placard. He collects it, and becomes a State Actor. The State Actors all gather, with some distance between them, in a designated region. (One might break down the region into sub-regions, for subsets of the states, if one wants.) </P><P>The Recorder gives the State Pointer to the State Actor holding the placard for the initial state. </P><P>The Recorder reads out the guidelines for the initial line of letters. Audience members come up and form a line, accordingly. As an example, the guidelines might say: </P><BLOCKQUOTE>The first player sits in the line and holds the "End" letter. The second player stands behind the first player. He holds a "Blank" and the Head. A number of players sit in the line behind the second player. They should each hold "0" or "1", as they choose. A person should sit after these players, and she holds a ";". Another number of players sit in in a row behind her. They also should each hold a "0" or "1". </BLOCKQUOTE><P>The Recorder writes down the sequence of letters in the initial set up. This step is optional. </P><P>Play can now commence. Play consists of a sequence of clock cycles. </P><B>3.2 A Clock Cycle</B><P>The player holding the Head commences a clock cycle. This player calls out the letter he is holding. </P><P>The state actor holding the State Pointer now plays. He looks at his rules and finds the rule corresponding to the letter that has been called out. Each rule has two parts. The first part is either a letter from the alphabet or the word "Forward" or the word "Backward". The second part is the name of a state. That state could be the label on the state placard that this State Actor is holding. Or it could be another state. </P><P>If the State Actor calls out a letter, an audience member comes up. He selects that letter from leftover letters in the initial set of equipment. He replaces the player holding the Head in the line. And that player hands the new player the Head. </P><P>If the State Actor calls out Forward, the player holding the Head hands it to the player holding a letter in front of him and sits down. The player now holding the Head stands up. There would be no such player if the player holding the Head at the start of the cycle is standing at the front of the line. In this case, an audience member picks up a "Blank" from the leftover set of equipment. That player accepts the Head and stands at the front of the line. </P><P>If the State Actor calls out Backward, the player holding the Head hands it to the player holding a letter behind him and sits down. As you might expect, that player now holding the Head stands up. This step might also result in a new player coming up from the audience and joining the line. And this new player would join the line at the back. </P><P>The State Actor holding the State Pointer now calls out the state listed on the second part of the rule he is executing. If that state is not the state listed on his state placard, he hands the State Pointer to the appropriate State Actor. </P><P>The Recorder writes down the new state that the State Pointer has now transitioned to. (This step is optional.) </P><B>3.3 Ending the Game</B><P>The game ends either when the players become convinced it could go on forever, or it ends when a State Actor holding a placard for a halting state receives the State Pointer. If the game ends in a halting state, the State Actor reads the corresponding phrase from the state placard. That phrase might be something like: </P><BLOCKQUOTE>You have been a Turing machine computing the sum of two non-negative integers, written in binary. </BLOCKQUOTE><P>Or it could be: </P><BLOCKQUOTE>You had at least one unmatched parentheses in your initial line. </BLOCKQUOTE><P>If you want, the Recorder could have more audience members come up to recreate the initial line. You can then review, if you like, the computation. For example, you might check that the sum of the two numbers separated by a comma in the initial line up is equal to the number now represented by the final line up on the stage. </P><B>4.0 Much To Do</B><P>Obviously, much would need to be done to flesh this out. In particular, equipment sets need to be constructed. Some choices to think about: </P><UL><LI>Would one want to include an equipment set in which the simulated Turing machine does not terminate for some initial line of letters? Or would one want to, at least for the first performance, only have rules that are guaranteed termination for all (valid?) inputs?</LI><LI>Might one want to emulate automata for languages lower down on the Chomsky hierarchy? For example, one might create a stack to be pushed and popped before the start of the line. Here I envision that a subset of the states specify subroutines. And the State Actors defining these subroutines might be grouped separately from the other actors.</LI><LI>Would one want to share alphabets among more than one equipment set? Maybe all six sets should have the same alphabet.</LI><LI>How would one describe the initial line up for a Turing machine that is to decide or semi-decide whether a given string is in a given language? The specification of a grammar for generating a string can be quite confusing to beginners.</LI><LI>I am thinking that one would not want to create rules for a universal Turing machine. Even some of the suggestions above might be too long to play.</LI></UL><P>An interesting variation would be to simulate a non-deterministic Turing machine. For some clock cycles, the line would be duplicated. And one would introduce another Head and State Pointer. </P><B>5.0 Instruction and Theatrics</B><P>This activity could serve pedagogical purposes. Suppose the players are different cohorts of students. Could the older students be directed to write the rules for some other computation at the next meeting? Could a set of recursive functions be built up over many meetings? Maybe one would end up with a group engaging in real-time debugging in a joint activity. </P><P>One could set up an accompanying talk or lecture. Many topics could be broached: The Church-Turing thesis and universality, uncomputable functions and the halting problem, computational complexity and the question of whether P equals NP, linguistics and the Chomsky hierarchy, etc. Or one might talk about the British secret service and reading the Nazi's mail. I guess there is both a Broadway play and a movie to go along with this activity. </P><P>One could introduce some sophistication in showmanship, depending on where this concept is instantiated. I like the idea of the alphabet players wearing different colored shirts, with each color corresponding to a character. Zero could be red, and one could be green. Blanks would be a neutral color, such as white. The State Actors could be in a dim area, with a spotlight serving as the State Pointer. The State Actors or the letter holders could be members of an orchestra, with some tune being played for every state transition or invoked rule. At termination, the entire derivation written down by the Recorder could be run-through. I imagine it would be difficult to design a set of rules that results in an interesting tune. At any rate, I guess the interests of an observing mathematician, the participants, and a theatergoer would be in tension. </P><P>I hope if somebody was to try this project, they would give me appropriate acknowledgement. </P><B>Reference</B><UL><LI>Lou Fisher (1975). "Nobody Named Gallix", <I>The Magazine of Fantasy and Science Fiction</I> (Jan.): pp. 98-109.</LI><LI>Andrew Hodges (1983). <I>Alan Turing: The Enigma</I>, Princeton University Press.</LI><LI>HarryR. Lewis and Christos H. Papadimitriou (1998). <I>Elements of the Theory of Computation</I>, 2nd edition. Prentice Hall.</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-47279410192700315972016-04-13T15:32:00.000-04:002016-04-18T06:58:21.378-04:00Math Is Power<B>1.0 Introduction</B><P>A common type of post in this blog is the presentation of concrete numerical examples in economics. Sometimes I present examples to illustrate some principle. But usually I try to find examples that are counter-intuitive or perverse, at least from the perspective of economics as mainstream economists often misteach it. </P><P>Voting games provide an arena where one can find surprising results in political science. I am thinking specifically of power indices. In this post, I try to explain two of the most widely used power indices by means of an example. </P><B>2.0 Me and My Aunt: A Voting Game</B><P>For purposes of exposition, I consider a specific game, called <I>Me and My Aunt</I>. There are four players in this version of the game, represented by elements of the set: </P><BLOCKQUOTE><I>P</I> = The set of players = {0, 1, 2, 3} </BLOCKQUOTE><P>Out of respect, the first player gets two votes, while all other players get a vote each (Table 1). A coalition, <I>S</I>, is a set of players. That is, a coalition is a subset of <I>P</I>. A coalition passes a resolution if it has a majority of votes. Since there are four players, one of who has two votes, the total number of votes is five. So a majority, in this game of weighted voting, is three votes. </P><table align="CENTER" border=""><tbody><CAPTION><b>Table 1: Players and Their Votes</b></CAPTION><TR align="CENTER"><TD><B>Players</B></TD><TD><B>Votes</B></TD></TR><TR align="CENTER"><TD>0 (Aunt)</TD><TD>2</TD></TR><TR align="CENTER"><TD>1 (Me)</TD><TD>1</TD></TR><TR align="CENTER"><TD>2</TD><TD>1</TD></TR><TR align="CENTER"><TD>3</TD><TD>1</TD></TR></tbody></table><P>One needs to specify the payoff to each coalition to complete the definition, in characteristic function form, of this game. The <A HREF="http://robertvienneau.blogspot.com/2008/12/dont-say-there-must-be-something-common.html">characteristic function</A>, <I>v</I>(<I>S</I>) maps the set of all subsets of <I>P</I> to the set {0, 1}. If the players in <I>S</I> have three or more votes,<I>v</I>(<I>S</I>) is 1. Otherwise, it is 0. That is, a winning coalition gains a payoff of one to share among its members. </P><B>3.0 The Penrose-Banzhaf Power Index</B><P>Power for a player, in this mathematical approach, is the ability to be the decisive member of a coalition. If, for a large number of coalitions, you being in or out of a coalition determines whether or not that coalition can pass a resolution, you have a lot of power. Correspondingly, if the members of most coalitions do not care whether you join, because your presence has no influence on whether or not they can put their agenda into effect, you have little power. </P><P>The Penrose-Banzhaf power index is one (of many) attempts to quantify this idea. Table 2 lists all 16 coalitions for the voting game under consideration. (The number of coalitions is the sum of a row in Pascal's triangle.) The second column in Table 2 specifies the value for the characteristic function for that coalition. Equivalently, the third column notes which eight coalitions are winning coalitions, and which eight are losing. The last two columns are useful for tallying up counts needed for the Penrose-Banzhaf index. </P><table align="CENTER" border=""><tbody><CAPTION><b>Table 2: Calculations for Penrose-Banzhaf Power Index</b></CAPTION><TR align="CENTER"><TD ROWSPAN="2"><B>Coalition</B></TD><TD ROWSPAN="2"><B>Characteristic<BR>Function</B></TD><TD ROWSPAN="2"><B>Winning<BR>or Losing</B></TD><TD COLSPAN="2"><B>Player</B></TD></TR><TR align="CENTER"><TD><B>Aunt (0)</B></TD><TD><B>Me (1)</B></TD></TR><TR align="CENTER"><TD>{}</TD><TD><I>v</I>( {} ) = 0</TD><TD>Losing</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>{0}</TD><TD><I>v</I>( {0} ) = 0</TD><TD>Losing</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>{1}</TD><TD><I>v</I>( {1} ) = 0</TD><TD>Losing</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>{2}</TD><TD><I>v</I>( {2} ) = 0</TD><TD>Losing</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>{3}</TD><TD><I>v</I>( {3} ) = 0</TD><TD>Losing</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>{0, 1}</TD><TD><I>v</I>( {0, 1} ) = 1</TD><TD>Winning</TD><TD>1</TD><TD>1</TD></TR><TR align="CENTER"><TD>{0, 2}</TD><TD><I>v</I>( {0, 2} ) = 1</TD><TD>Winning</TD><TD>1</TD><TD>0</TD></TR><TR align="CENTER"><TD>{0, 3}</TD><TD><I>v</I>( {0, 3} ) = 1</TD><TD>Winning</TD><TD>1</TD><TD>0</TD></TR><TR align="CENTER"><TD>{1, 2}</TD><TD><I>v</I>( {1, 2} ) = 0</TD><TD>Losing</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>{1, 3}</TD><TD><I>v</I>( {1, 3} ) = 0</TD><TD>Losing</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>{2, 3}</TD><TD><I>v</I>( {2, 3} ) = 0</TD><TD>Losing</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>{0, 1, 2}</TD><TD><I>v</I>( {0, 1, 2} ) = 1</TD><TD>Winning</TD><TD>1</TD><TD>0</TD></TR><TR align="CENTER"><TD>{0, 1, 3}</TD><TD><I>v</I>( {0, 1, 3} ) = 1</TD><TD>Winning</TD><TD>1</TD><TD>0</TD></TR><TR align="CENTER"><TD>{0, 2, 3}</TD><TD><I>v</I>( {0, 2, 3} ) = 1</TD><TD>Winning</TD><TD>1</TD><TD>0</TD></TR><TR align="CENTER"><TD>{1, 2, 3}</TD><TD><I>v</I>( {1, 2, 3} ) = 1</TD><TD>Winning</TD><TD>0</TD><TD>1</TD></TR><TR align="CENTER"><TD>{0, 1, 2, 3}</TD><TD><I>v</I>( {0, 1, 2, 3} ) = 1</TD><TD>Winning</TD><TD>0</TD><TD>0</TD></TR></tbody></table><P>The Penrose-Banzhaf index, &psi;(<I>i</I>) is calculated for each player <I>i</I>. It is defined, for a given player, to be the ratio of the number of winning coalitions in which that player is decisive to the total number of coalitions, winning or losing. A player is decisive for a coalition if: </P><UL><LI>The coalition is a winning coalition.</LI><LI>The removal of the player from the coalition converts it to a losing coalition.</LI></UL><P>From the table above, one can see that player 0 is decisive for six coalitions, while player 1 is decisive for only two coalitions. Hence, the Penrose-Banzhaf index for "my aunt" is: </P><BLOCKQUOTE>&psi;(0) = 6/16 = 3/8 </BLOCKQUOTE><P>By symmetry, the index values for players 2 and 3 are the same as the value for player 1: </P><BLOCKQUOTE>&psi;(1) = &psi;(2) = &psi;(3) = 2/16 = 1/8 </BLOCKQUOTE><P>More than one player can be decisive for a winning coalition. No need exists for the Penrose-Banzhaf index to sum up to one. How much one's vote is weighted does not bear a simple relationship to how much power one has. Also note that the definition of this power index is not confined to simple majority games. Power indices can be calculated for voting games in which a super-majority is required to pass a measure. For example, in the United States Senate, 60 senators are needed to end a filibuster. </P><B>4.0 The Shapley-Shubik Power Index</B><P>The Shapley-Shubik power index is an application of the calculation of the Shapley value to voting games. The Shapley value applies to cooperative games, in general. For its use as a measure of power in voting games, it matters in which order players enter a coalition. Accordingly, Table 3 lists all 24 permutations of all four players in the voting game being analyzed. </P><table align="CENTER" border=""><tbody><CAPTION><b>Table 3: Calculations for the Shapley-Shubik Power Index</b></CAPTION><TR align="CENTER"><TD ROWSPAN="2"><B>Permutation</B></TD><TD COLSPAN="2"><B>Player</B></TD></TR><TR align="CENTER"><TD><B>Aunt (0)</B></TD><TD><B>Me (1)</B></TD></TR><TR align="CENTER"><TD>(0, 1, 2, 3)</TD><TD><I>v</I>( {0} ) - <I>v</I>( {} ) = 0</TD><TD><I>v</I>( {0, 1} ) - <I>v</I>( {0} ) = 1</TD></TR><TR align="CENTER"><TD>(0, 1, 3, 2)</TD><TD><I>v</I>( {0} ) - <I>v</I>( {} ) = 0</TD><TD><I>v</I>( {0, 1} ) - <I>v</I>( {0} ) = 1</TD></TR><TR align="CENTER"><TD>(0, 2, 1, 3)</TD><TD><I>v</I>( {0} ) - <I>v</I>( {} ) = 0</TD><TD><I>v</I>( {0, 1, 2} )<BR> - <I>v</I>( {0, 2} ) = 0</TD></TR><TR align="CENTER"><TD>(0, 2, 3, 1)</TD><TD><I>v</I>( {0} ) - <I>v</I>( {} ) = 0</TD><TD><I>v</I>( {0, 1, 2, 3} )<BR>- <I>v</I>( {0, 2, 3} ) = 0</TD></TR><TR align="CENTER"><TD>(0, 3, 1, 2)</TD><TD><I>v</I>( {0} ) - <I>v</I>( {} ) = 0</TD><TD><I>v</I>( {0, 1, 3} )<BR>- <I>v</I>( {0, 3} ) = 0</TD></TR><TR align="CENTER"><TD>(0, 3, 2, 1)</TD><TD><I>v</I>( {0} ) - <I>v</I>( {} ) = 0</TD><TD><I>v</I>( {0, 1, 2, 3} )<BR>- <I>v</I>( {0, 2, 3} ) = 0</TD></TR><TR align="CENTER"><TD>(1, 0, 2, 3)</TD><TD><I>v</I>( {0, 1} )<BR>- <I>v</I>( {1} ) = 1</TD><TD><I>v</I>( {1} ) - <I>v</I>( {} ) = 0</TD></TR><TR align="CENTER"><TD>(1, 0, 3, 2)</TD><TD><I>v</I>( {0, 1} )<BR>- <I>v</I>( {1} ) = 1</TD><TD><I>v</I>( {1} ) - <I>v</I>( {} ) = 0</TD></TR><TR align="CENTER"><TD>(1, 2, 0, 3)</TD><TD><I>v</I>( {0, 1, 2} )<BR>- <I>v</I>( {1, 2} ) = 1</TD><TD><I>v</I>( {1} ) - <I>v</I>( {} ) = 0</TD></TR><TR align="CENTER"><TD>(1, 2, 3, 0)</TD><TD><I>v</I>( {0, 1, 2, 3} )<BR>- <I>v</I>( {1, 2, 3} ) = 0</TD><TD><I>v</I>( {1} ) - <I>v</I>( {} ) = 0</TD></TR><TR align="CENTER"><TD>(1, 3, 0, 2)</TD><TD><I>v</I>( {0, 1, 3} )<BR>- <I>v</I>( {1, 3} ) = 1</TD><TD><I>v</I>( {1} ) - <I>v</I>( {} ) = 0</TD></TR><TR align="CENTER"><TD>(1, 3, 2, 0)</TD><TD><I>v</I>( {0, 1, 2, 3} )<BR>- <I>v</I>( {1, 2, 3} ) = 0</TD><TD><I>v</I>( {1} ) - <I>v</I>( {} ) = 0</TD></TR><TR align="CENTER"><TD>(2, 0, 1, 3)</TD><TD><I>v</I>( {0, 2} )<BR>- <I>v</I>( {2} ) = 1</TD><TD><I>v</I>( {0, 1, 2} )<BR>- <I>v</I>( {0, 2} ) = 0</TD></TR><TR align="CENTER"><TD>(2, 0, 3, 1)</TD><TD><I>v</I>( {0, 2} )<BR>- <I>v</I>( {2} ) = 1</TD><TD><I>v</I>( {0, 1, 2, 3} )<BR>- <I>v</I>( {0, 2, 3} ) = 0</TD></TR><TR align="CENTER"><TD>(2, 1, 0, 3)</TD><TD><I>v</I>( {0, 1, 2} )<BR>- <I>v</I>( {1, 2} ) = 1</TD><TD><I>v</I>( {1, 2} ) - <I>v</I>( {2} ) = 0</TD></TR><TR align="CENTER"><TD>(2, 1, 3, 0)</TD><TD><I>v</I>( {0, 1, 2, 3} )<BR>- <I>v</I>( {1, 2, 3} ) = 0</TD><TD><I>v</I>( {1, 2} ) - <I>v</I>( {2} ) = 0</TD></TR><TR align="CENTER"><TD>(2, 3, 0, 1)</TD><TD><I>v</I>( {0, 2, 3} )<BR>- <I>v</I>( {2, 3} ) = 1</TD><TD><I>v</I>( {0, 1, 2, 3} )<BR>- <I>v</I>( {0, 2, 3} ) = 0</TD></TR><TR align="CENTER"><TD>(2, 3, 1, 0)</TD><TD><I>v</I>( {0, 1, 2, 3} )<BR>- <I>v</I>( {1, 2, 3} ) = 0</TD><TD><I>v</I>( {1, 2, 3} )<BR>- <I>v</I>( {2, 3} ) = 1</TD></TR><TR align="CENTER"><TD>(3, 0, 1, 2)</TD><TD><I>v</I>( {0, 3} )<BR>- <I>v</I>( {3} ) = 1</TD><TD><I>v</I>( {0, 1, 3} )<BR>- <I>v</I>( {0, 3} ) = 0</TD></TR><TR align="CENTER"><TD>(3, 0, 2, 1)</TD><TD><I>v</I>( {0, 3} )<BR>- <I>v</I>( {3} ) = 1</TD><TD><I>v</I>( {0, 1, 2, 3} )<BR>- <I>v</I>( {0, 2, 3} ) = 0</TD></TR><TR align="CENTER"><TD>(3, 1, 0, 2)</TD><TD><I>v</I>( {0, 1, 3} )<BR>- <I>v</I>( {1, 3} ) = 1</TD><TD><I>v</I>( {1, 3} ) - <I>v</I>( {3} ) = 0</TD></TR><TR align="CENTER"><TD>(3, 1, 2, 0)</TD><TD><I>v</I>( {0, 1, 2, 3} )<BR>- <I>v</I>( {1, 2, 3} ) = 0</TD><TD><I>v</I>( {1, 3} ) - <I>v</I>( {3} ) = 0</TD></TR><TR align="CENTER"><TD>(3, 2, 0, 1)</TD><TD><I>v</I>( {0, 2, 3} )<BR>- <I>v</I>( {2, 3} ) = 1</TD><TD><I>v</I>( {0, 1, 2, 3} )<BR>- <I>v</I>( {0, 2, 3} ) = 0</TD></TR><TR align="CENTER"><TD>(3, 2, 1, 0)</TD><TD><I>v</I>( {0, 1, 2, 3} )<BR>- <I>v</I>( {1, 2, 3} ) = 0</TD><TD><I>v</I>( {1, 2, 3} )<BR>- <I>v</I>( {2, 3} ) = 1</TD></TR></tbody></table><P>Table 3 shows some initially confusing calculations in the last two columns, where each of these columns is defined for a given player. Suppose a player and a permutation are defined. For that permutation, let the set <I>S</I><SUB>&pi;, <I>i</I></SUB> contain those players in the permutation &pi; to the left of the given player <I>i</I>. The difference, in the last two columns, is the following, for <I>i</I> equal to 0 and to 1, respectively: </P><BLOCKQUOTE><I>v</I>(<I>S</I><SUB>&pi;, <I>i</I></SUB> &#8746; {<I>i</I>}) - <I>v</I>(<I>S</I><SUB>&pi;, <I>i</I></SUB>) </BLOCKQUOTE><P>The Shapley-Shubik power index, for a player, is the ratio of a sum to the number of permutations of players. And that sum is calculated for each player, as the sum over all permutations, of the above difference in the value of the value of the characteristic function. </P><P>If I understand correctly, given a permutation, the above difference can only take on values of 0 or 1. And it will only be 1 for one player, where that player determines whether the formation of the coalition in the order given will be a winning coalition. As a consequence, the Shapley-Shubik power index is guaranteed to sum over players to unity. In this case, power is a fixed amount, with each player being measured as having a defined proportion of that power. </P><B>5.0 Both Power Indices</B><P>The above has stepped through the calculation of two power indices, for all players, in a given game. Table 4 lists their values, as well as a normalization of the Penrose-Banzhauf power index such that the sum of the power, over all players, is unity. (I gather that the Penrose-Banzhauf index and the normalized index do not have the same properties.) As one might expect from the definition of the game, "my aunt" has more power than "me" in this game. </P><table align="CENTER" border=""><tbody><CAPTION><b>Table 4: The Penrose-Banzhaf and Shapley-Shubik Power Indices</b></CAPTION><TR align="CENTER"><TD ROWSPAN="2"><B>Player</B></TD><TD COLSPAN="2"><B>Penrose-Banzhaf Power Index</B></TD><TD ROWSPAN="2"><B>Shapley-Shubik<BR>Power Index</B></TD></TR><TR align="CENTER"><TD><B>Index</B></TD><TD><B>Normalized</B></TD></TR><TR align="CENTER"><TD>0</TD><TD>6/16 = 3/8</TD><TD>6/12 = 1/2</TD><TD>12/24 = 1/2</TD></TR><TR align="CENTER"><TD>1</TD><TD>2/16 = 1/8</TD><TD>2/12 = 1/6</TD><TD>4/24 = 1/6</TD></TR><TR align="CENTER"><TD>2</TD><TD>2/16 = 1/8</TD><TD>2/12 = 1/6</TD><TD>4/24 = 1/6</TD></TR><TR align="CENTER"><TD>3</TD><TD>2/16 = 1/8</TD><TD>2/12 = 1/6</TD><TD>4/24 = 1/6</TD></TR></tbody></table><P>In many voting games, the normalized Penrose-Banzhauf and Shapley-Shubik power indices are not identical for all players. In fact, suppose the rules for the above variation of <I>Me and my Aunt</I> voting game are varied. Suppose now that four votes - a supermajority - are needed to carry a motion. The normalized Penrose-Banzhaf index for player 0 becomes 1/3, while each of the other players have a normalized Penrose-Banzhaf index of 2/9. Interestingly enough, the Shapley-Shubik indices for the players do not change, if I have calculated correctly. But the values assigned to rows in Table 3 do sometimes vary. Anyways, that one tweak of the rules results in different power indices, depending on which method one adopts. A more interesting example would be one in which the rankings vary among power indices. </P><P>Other power indices, albeit less common, do exist. Which one is most widely applicable? I would think that mainstream economists, given game theory and marginalism, would tend to prefer the Shapley-Shubik power index. Felsenthal and Machover (2004) seem to be widely recognized experts on measures of voting power, and they have come to prefer the Penrose-Banzhaf index over the Shapley-Shubik index. </P><B>6.0 Where To Go From Here</B><P>I have described above a couple of power indices in voting games. As I understand it, many have tried to write down reasonable axioms that characterize power indices. One challenge is to specify a set of axioms such that your preferred power index is the only one that satisfies them. But, as I understand it, some sets of reasonable axioms are open insofar as more than one power index would satisfy them. I seem to recall a theorem that one could create a power index for a reasonable set of axioms such that whichever player you want in a voting game is the most powerful. Apparently, a connection can be drawn between a power index and a voting procedure. And Donald Saari <A HREF="http://robertvienneau.blogspot.com/2008/11/militant-voting.html">boasts</A> that he could create an apparently fair voting procedure that would result in whatever candidate you like being elected. </P><P>I gather that many examples of voting games have been presented in which apparently paradoxical or perverse results arise. And these do not seem to be merely theoretical results. Can I find some such examples? Perhaps, I should look here at some of <A HREF="http://robertvienneau.blogspot.com/2011/02/daron-acemoglu.html">Daron Acemoglu's</A> work. </P><P>I am aware of three types of examples to look for. One is that of a dummy. A dummy is a player that, under the weights and the rule for how many votes are needed for passage, can never be decisive in a coalition. Whether this player drops out or joins a coalition can never change whether or not a resolution is passed, even though the player has a positive weight. A second odd possibility arises as the consequence of adding a new member to the electorate: </P><BLOCKQUOTE>"...power of a weighted voting body may increase, rather than decrease, when new members are added to the original body." -- Steven J. Brams and Paul J. Affuso (1976). </BLOCKQUOTE><P>A third odd possibility apparently can arise on a council when one district annexes another. Suppose, the district annexing the other consequently increases the weight of its vote accordingly. One might think a greater weight leads to more power. But, in certain cases, the normalized Penrose-Banzhaf index can decrease. </P><P>The above calculations for the Penrose-Banzhaf and Shapley-Shubik power indices treat all coalitions or permutations, respectively, as equally likely to arise. Empirically, this does not seem to be true. And this has an impact on how one might measure power. For example, since voting is unweighted on the Supreme Court of the United States, all justices might be thought to be equally powerful. But, because of the formation of well-defined blocks, Anthony Kennedy was often described as being particularly powerful in deciding court decisions, at least when Antonin Scalia was still alive. So empirically, one might include some assessment of the affinities of the players for one another and, thus, some influence on the probabilities of each coalition forming. This will have consequences on the calculation of power indices. But why stop there? In the United States these days, <A HREF="http://robertvienneau.blogspot.com/2013/04/political-elites-bowing-down-before.html">politicians</A> only seem to <A HREF="http://robertvienneau.blogspot.com/2012/09/your-opinion-does-not-matter.html">represent</A> the most <A HREF="http://robertvienneau.blogspot.com/2013/05/our-rulers-do-not-know-why-they-dislike.html">wealthy</A>. </P><P><B>Update:</B> This <A HREF="http://homepages.warwick.ac.uk/~ecaae/#Progam_List">page</A>, from the University of Warwick, has links to utilities for calculating various power indices. </P><B>References</B><UL><LI>Steven J. Brams and Paul J. Affuso (1976). Power and Size: A New Paradox, <I>Theory and Decision</I>. V. 7, Iss. 1 (Feb.): pp. 29-56.</LI><LI>Dan S. Felsenthal and Mosh&eacute; Machover (2004). Voting Power Measurement: A Story of Misreinvention, London Scool of Economics and Political Science</LI><LI>Andrew Gelman, Jonathan N. Katz, and Joseph Bafumi (2004). Standard Voting Power Indexes Do Not Work: An Empirical Analysis, <I>B. J. Pol. S.</I>. V. 34: pp. 657-674.</LI><LI>Guillermo Owen (1971) Political Games, <I>Naval Research Logistics Quarterly</I>. V. 18, Iss. 3 (Sep.): pp. 345-355.</LI><LI>Donald G. Saari and Katri K. Sieberg (1999). Some Surprising Properties of Power Indices.</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-21179391124817521102016-04-11T13:00:00.000-04:002016-04-11T13:00:05.632-04:00Inane Responses To The Cambridge Capital Controversy<P>I consider the following views, if unqualified and without caveats, just silly: </P><UL><LI>The Cambridge Capital Controversy (CCC) was only attacking aggregate neoclassical theory.</LI><LI>The CCC is just a General Equilibrium argument, and it has been subsumed by General Equilibrium Theory. (Citing Mas Colell (1989) here does not help.)</LI><LI>The CCC does not have anything to say about partial, microeconomic models.</LI><LI>Perverse results, such as reswitching and capital-reversing, only arise in the special case of Leontief production functions. If you adopt widely used forms for production functions, the perverse results go away.</LI><LI>It is an empirical question whether non-perverse results follow from neoclassical assumptions. And nobody has ever found empirical examples of capital-reversing or reswitching.</LI><LI>Mainstream economists have moved on since the 1960s, and their models these days are not susceptible to the Cambridge critique.</LI></UL><P>I would think that one could not get such ideas published in any respectable journal. On the other hand, Paul Romer did get his <A HREF="https://criticalfinance.org/2016/03/27/economics-science-or-politics-a-reply-to-kay-and-romer/">ignorance</A> about Joan Robinson into the <I>American Economic Review</I></P><B>References</B><UL><LI>Andreu Mas-Colell (1989). Capital theory paradoxes: Anything goes. In <I>Joan Robinson and Modern Economic Theory</I> (ed. by George R. Feiwel), Macmillan.</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-38947436794814467082016-03-19T16:54:00.000-04:002016-03-22T07:04:07.010-04:00Post Keynesianism From The Outside<P>Post Keynesian economics has been under development for about three quarters of a century now. Academics in countries around the world have made contributions to the theory and to its application. And they have participated in many common practices of academics, including economists<SUP>1</SUP>. </P><P>Post Keynesians have written and published papers in peer-reviewed journals. Over this time-span, such journals include widely referenced mainstream journals, such as the <I>American Economic Review</I>, the <I>Economic Journal</I>, the <I>Journal of Economic Literature</I>, the <I>Journal of Political Economy</I>, and the <I>Quarterly Journal of Economics</I><SUP>2</SUP>. Lately, certain specialized journals have proven more sympathetic to publishing Post Keynesians. Such journals include, for example, the <I>Cambridge Journal of Economics</I>, the <I>Journal of Post Keynesian Economics</I>, <I>Kyklos</I>, and the <I>Review of Political Economy</I>. The list suggests two other activities: the founding and editing of journals. As I understand it, Joan Robinson, among other economists now thought of as Post Keynesian, participated in the founding of the <I>Review of Economic Studies</I>, while the <I>Review of Keynesian Economics</I> is a more recent academic journal with an analogous start. The <I>Banca Nazionale del Lavoro Quarterly Review</I>, the <I>Canadian Journal of Economics</I>, and <I>Metroeconomica</I> are some journals, while not being specifically heterodox, I guess, had Post Keynesians as editors for some time<SUP>3, 4</SUP>. </P><P>Participation in professional societies, as officers, as organizers of conferences and conference sessions, and as presenters at conferences, provides another typical venue for academic activities. Naturally, Post Keynesians have performed such activities. For example, John Kenneth Galbraith was president of the American Economic Society, and the annual meeting of the Allied Social Sciences Association (ASSA), held in conjunction with the American Economics Association, regularly holds sessions dedicated to Post Keynesian topics. Recently, heterodox economics have become interested in pluralism and how their concerns overlap. These concerns have been reflected in much work in many professional societies relating to heterodox economics. </P><P>I began this article with journal publications because economics has become less focused on books and more focused on journal publications during the period in which Post Keynesianism grew. But during this period, Post Keynesians have also made original contributions in books published by prestige university and academic presses. I think, for example, of presses associated with Cambridge, Columbia, and Harvard, to pick some examples at random<SUP>5</SUP>. </P><P>After decades of work, a need will arise to introduce others to it. And Post Keynesians have addressed this need with anthologies of classic papers, introductory <A HREF="http://robertvienneau.blogspot.com/2014/05/paradigming-is-easy.html">works</A> for other economists, and <A HREF="http://robertvienneau.blogspot.com/2006/08/textbooks-for-teaching-non.html">textbooks</A>. One can also find Post Keynesians editing, or participating in the development, of standard reference works<SUP>6</SUP>. </P><P>On a more local level, Post Keynesian economists have participated in the governance of economic departments around the world<SUP>7</SUP>. And they have provided governments with advice many a time, from within and without<SUP>8</SUP>. </P><P>I have deliberately not written about substantial Post Keynesian ideas in this post. If one is aware of the history mentioned in this post, even if one had never been exposed to Post Keynesian ideas, one must conclude Post Keynesian theory is much like any other set of academic ideas. One would have difficulty in seeing how academics could justify dismissing these ideas without engaging with them Likewise, one might wonder how, perhaps, those aspiring to be professional economists might not even be exposed to Post Keynesianism in gaining a post graduate degree. Yet, apparently, such a happenstance seems to be not at all rare among mainstream economists. </P><B>Footnotes</B><OL><LI>I recognize my post is biased towards the English language. It is also quite impressionistic and selective. I am taking Sraffians as Post Keynesians for the purposes of this post.</LI><LI>Major contributions to the Cambridge Capital Controversy are to be found in these journals.</LI><LI>For example, Paolo Sylos Labini for the <I>Banca Nazionale del Lavoro Quarterly Review</I>, Athanasios Asimakopulos for the <I>Canadian Journal of Economics</I>, and Neri Salvadori for <I>Metroeconomica</I>.</LI><LI>For what it is worth, I am published in the <I>Manchester School</I>.</LI><LI>How would one characterize Edward Elgar and Routledge, for example?</LI><LI>The first edition of the <I>New Palgrave</I> is an obvious example.</LI><LI>Economics at Cambridge University is an obvious case. Albert Eichner chairing the Rutgers economics department is another case.</LI><LI> Examples include Nicholas Kaldor's work with the Radcliffe Committee, John Kenneth Galbraith giving advice to John F. Kennedy, the advocacy of Tax-based Income Policies (TIPs) in the 1970s to fight stagflation, and policy suggestions associated with Modern Monetary Theory (MMT).</LI></OL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com2tag:blogger.com,1999:blog-26706564.post-90437476509717743982016-03-02T08:04:00.000-05:002016-03-02T08:04:04.692-05:00Romer And Romer Stumble<P>A <A HREF="http://billmoyers.com/story/the-battle-over-reagans-economic-plan/">debate</A> <A HREF="http://www.truth-out.org/news/item/34978-the-dominant-media-left-leaning-economists-and-the-illusion-of-consensus">has</A> <A HREF="http://big.assets.huffingtonpost.com/ResponsetoCEA.pdf">recently</A> <A HREF="http://jwmason.org/slackwire/plausibility/">arisen</A> <A HREF="http://econbrowser.com/archives/2016/02/what-is-the-assumed-output-gap-in-the-friedman-projections">about</A> Gerald Friedman's <A HREF="http://www.dollarsandsense.org/What-would-Sanders-do-013016.pdf">analysis</A> of Bernie Sanders' proposed economic program. In a welcome turn of events, two defenders of the establishment, Christina and David Romer, finally offer some substance, instance of just relying on their authority as Very Serious People. </P><P>In this post, I ignore most of the substance of the argument. I want to focus on three errors I find in this passage: </P><BLOCKQUOTE>"Potentially more worrisome are the extensive interventions in the labor market. The experiences of many European countries from the 1970s to today show that an overly regulated labor market can have severe consequences for normal unemployment. There are strong arguments for raising the minimum wage; and over the range observed historically in the United States, the short-run employment effects of moderate increases appear negligible. But doubling the minimum wage nationwide, adding new requirements for employer-funded paid vacations and sick leave, and increasing payroll taxes substantially would take us into uncharted waters. Obviously, these changes would not bring the United States all the way to levels of labor market regulation of many European countries in the 1970s. But they are large enough that one can reasonably fear that they could have a noticeable impact on capacity growth." -- Christina D. Romer and David H. Romer, <A HREF="http://ineteconomics.org/uploads/general/romer-and-romer-evaluation-of-friedman1.pdf">Senator Sander's Proposed Policies and Economic Growth</A> (5 February 2016) p. 10-11. </BLOCKQUOTE><P>First, the reference to "interventions in the labor market" and an "overly regulated labor market" imposes a false dichotomy. An unregulated labor market cannot exist. Certainly this is so in an advanced capitalist economy. Possible choices are among sets of regulations and norms, not among intervention or not. Calling one set of regulations an example of government non-intervention is to disguise taking a side under obfuscatory verbiage. </P><P>Second, Romer and Romer presuppose a consensus about the empirical effects of different regulations on the labor market in Europe and the United States that I do not think exists. If I wanted to find empirically based arguments countering Romer and Romer's claim, I would look through back issues of the <I>Cambridge Journal of Economics</I>. Perhaps at <A HREF="http://cje.oxfordjournals.org/content/39/2/467.abstract">least</A> <A HREF="https://cje.oxfordjournals.org/content/37/4/845">one</A> <A HREF="https://cje.oxfordjournals.org/content/35/2/437.abstract?sid=21976fe7-e9d2-4a61-9793-e674e80ab7f3">of</A> <A HREF="https://cje.oxfordjournals.org/content/33/1/51.abstract?sid=21976fe7-e9d2-4a61-9793-e674e80ab7f3">these</A> <A HREF="https://cje.oxfordjournals.org/content/27/1/123.abstract?sid=21976fe7-e9d2-4a61-9793-e674e80ab7f3">articles</A> might be helpful. </P><P>Third, Romer and Romer suggest that, given the set of regulations they like to think of as government non-intervention, markets for labor and goods would have a tendency to clear. Otherwise, economic growth would be jeopardized. No theoretical foundation exists for thinking so. </P><P>Even the best mainstream economists seem incapable of writing ten pages without spouting ideological claptrap and propagating silly errors exposed more than half a century ago. Something seems terribly wrong with economics profession. </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-50803031409156361302016-02-29T08:03:00.000-05:002016-02-29T08:03:03.523-05:00Conservatism According to Corey Robin<P>I have been re-reading Corey Robin's <I>The Reactionary Mind</I>. According to this book, defending arbitrary hierarchies is the first priority among conservatives. They believe: </P><UL><LI>Workers should obey their masters.</LI><LI>Wives should obey their husbands.</LI><LI>Downtrodden ethnic groups should obey socially privileged ethnic groups.</LI><LI>The laity should obey priests.</LI><LI>The non-affluent should show proper deference towards those with great wealth (who could never be malefactors).</LI></UL><P>These hierarchies have implications for daily lives, not just political rule. For the right, liberty is liberty for the rulers to do as they will, not for those who suffer what they must. </P><P>I deliberately do not write about slavery. According to Robin, conservatism is literally reactionary. Conservatives defend hierarchies that are currently threatened or recently overthrown. They focus on restoring what was recently lost. Maybe this has something to do with widespread fear and resentment on the right. </P><P>Conservatives often do not have admiration for the rulers of the ancient regime. If those rulers were willing to do what needed to be done to preserve their power, the threats would never have gotten so far, and losses would not have been suffered. The conservative, unlike his popular and complimentary image, is willing to make radical changes so as to reconstruct society as it once was. This seems to go along with the awareness of some contemporary neoliberals that market societies are not natural formations, but must be constructed and maintained by state power. But is this aping of the left consistent with the conservative's encouragement of anti-intellectualism and stupidity? Perhaps the idea is that only an elite need understand the goal, while widespread ignorance among the masses can only help the cause. </P><P>The hierarchies that conservatives seek to defend or restore are not meritocracies, in the sense that those on top are expected to have superior intellect, wisdom, or morals. Rulers should demonstrate their fitness to rule by seizing what they can, in war or business. Maybe this has something to do with why many conservatives endorse the supposed "free market", without worrying about externalities, information asymmetries, transaction costs, or market power. One can also see here an echo of Friedrich Nietzsche's overman. </P><P>Much of the above comes from the introduction and first couple of chapters of Robin's book. Much of the rest consists of case studies of particular thinkers and polemicists. </P><B>Reference</B><UL><LI>Corey Robin (2013). <A HREF="http://www.amazon.com/The-Reactionary-Mind-Conservatism-Edmund/dp/0199959110"><I>The Reactionary Mind: Conservatism from Edmund Burke to Sarah Palin</I></A>, Oxford University Press.</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com1tag:blogger.com,1999:blog-26706564.post-66542433942826114432016-02-17T08:08:00.000-05:002016-02-17T08:08:02.217-05:00Classification of Finite Simple Groups: A Proved Theorem?<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><tr><td align="center"><a href="https://2.bp.blogspot.com/-2u-dgwXt7Mo/Vr3RzZOW5gI/AAAAAAAAAoc/rB6zXwFt2mc/s1600/D4Lattice.JPG" imageanchor="1"><img border="0" src="https://2.bp.blogspot.com/-2u-dgwXt7Mo/Vr3RzZOW5gI/AAAAAAAAAoc/rB6zXwFt2mc/s320/D4Lattice.JPG" /></a></td></tr><tr><td align="center"><b>Figure 1: Lattice Diagram for Group of Symmetries of the Square</b></td></tr></tbody></table><Blockquote>"I shall now mention something I obviously do not understand." - Ian Hacking (2014, p. 18) </BLOCKQUOTE><b>1.0 Introduction</b><P>This has nothing to do with economics. It is my attempt to get my mind around a place where I can get a glimmer of some exciting mathematics being done in my lifetime. </P><P>Mathematicians have stated a theorem for classifying finite simple groups. Whether they have proven this theorem is an intriguing question in the philosophy of mathematics. </P><P>A finite simple group is a group with a finite number of elements and no proper normal subgroup. This definition contains several technical terms. In this post, I try to explain these terms and the setting of the theorem for classifying simple groups. This preamble raises several questions: <P><ul><li>What is a group? A proper subgroup? A normal subgroup?</li><li>How can a finite, non-simple group be factored into a composition of simple groups?</li></ul><P>I try to clarify the answers to these questions by means of a lengthy example. You can probably find this better expressed elsewhere. In working this out, I relied heavily on Fraleigh's textbook, which is the only book in the references that I have read, albeit mostly in the second edition. </P><b>2.0 The Group of Symmetries of the Square</b><P>A <i>group</i> is a generalization, in some sense, of a multiplication table. Formally, it is a set with a binary operation, in which the binary operation satisfies three axioms. A <i>finite group</i> is a group in which the set contains a finite number of elements. </P><P>To illustrate, I consider the set of symmetries of the square (Figure 2). These eight elements of the set are like the numbers along the top and left side of a multiplication table. Each element is an operation that can be performed on a square, leaving the square superimposed on itself. Each operation is described in the right column of Figure 2. The third column provides a picture of the operation. The four vertices of the square are numbered so that one can see the result of the operation. The second column specifies each operation as a permutation of the numbered vertices. The first row in each permutation lists the vertices, while the second row shows which of the original vertices ends up in the place of each vertex. The first column introduces a notation for naming each operation. The remainder of this post is expressed in this notation. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><tr><td align="center"><a href="https://4.bp.blogspot.com/-MDXjdMdZx64/Vr3RrfNqLtI/AAAAAAAAAoY/BlP91-FWgVI/s1600/D4Definition.JPG" imageanchor="1"><img border="0" src="https://4.bp.blogspot.com/-MDXjdMdZx64/Vr3RrfNqLtI/AAAAAAAAAoY/BlP91-FWgVI/s320/D4Definition.JPG" /></a></td></tr><tr><td align="center"><b>Figure 2: Elements of a Group</b></td></tr></tbody></table><P>The group operation, *, is function composition. Let <i>a</i> and <i>b</i> be elements of the set {ρ<sub>0</sub>, ρ<sub>1</sub>, ρ<sub>2</sub>, ρ<sub>0</sub>, μ<sub>0</sub>, μ<sub>1</sub>, σ<sub>0</sub>, σ<sub>1</sub>}. The product <i>a</i>*<i>b</i> is defined to be the single operation that is equivalent to first performing the operation <i>a</i> on the square and then performing the operation <i>b</i> on the result. (Many textbooks define functional composition from right-to-left, instead.) Table 1 is the multiplication table for this group, under these definitions. For example, rotating a square 90 degrees clockwise twice is equivalent to rotating the square clockwise through 180 degrees. Thus:</P><BLOCKQUOTE>&rho;<SUB>1</SUB> * &rho;<SUB>1</SUB> = &rho;<SUB>2</SUB></BLOCKQUOTE><table align="CENTER" border=""><tbody><CAPTION><b>Table 1: The Group D<sub>4</sub></b></CAPTION><tr align="CENTER"><td><b>*</b></td><td><b>&rho;<sub>0</sub></b></td><td><b>&rho;<sub>1</sub></b></td><td><b>&rho;<sub>2</sub></b></td><td><b>&rho;<sub>3</sub></b></td><td><b>&mu;<sub>0</sub></b></td><td><b>&mu;<sub>1</sub></b></td><td><b>&sigma;<sub>0</sub></b></td><td><b>&sigma;<sub>1</sub></b></td></tr><tr align="CENTER"><td><b>&rho;<sub>0</sub></b></td><td>&rho;<sub>0</sub></td><td>&rho;<sub>1</sub></td><td>&rho;<sub>2</sub></td><td>&rho;<sub>3</sub></td><td>&mu;<sub>0</sub></td><td>&mu;<sub>1</sub></td><td>&sigma;<sub>0</sub></td><td>&sigma;<sub>1</sub></td></tr><tr align="CENTER"><td><b>&rho;<sub>1</sub></b></td><td>&rho;<sub>1</sub></td><td>&rho;<sub>2</sub></td><td>&rho;<sub>3</sub></td><td>&rho;<sub>0</sub></td><td>&sigma;<sub>0</sub></td><td>&sigma;<sub>1</sub></td><td>&mu;<sub>1</sub></td><td>&mu;<sub>0</sub></td></tr><tr align="CENTER"><td><b>&rho;<sub>2</sub></b></td><td>&rho;<sub>2</sub></td><td>&rho;<sub>3</sub></td><td>&rho;<sub>0</sub></td><td>&rho;<sub>1</sub></td><td>&mu;<sub>1</sub></td><td>&mu;<sub>0</sub></td><td>&sigma;<sub>1</sub></td><td>&sigma;<sub>0</sub></td></tr><tr align="CENTER"><td><b>&rho;<sub>3</sub></b></td><td>&rho;<sub>3</sub></td><td>&rho;<sub>0</sub></td><td>&rho;<sub>1</sub></td><td>&rho;<sub>2</sub></td><td>&sigma;<sub>1</sub></td><td>&sigma;<sub>0</sub></td><td>&mu;<sub>0</sub></td><td>&mu;<sub>1</sub></td></tr><tr align="CENTER"><td><b>&mu;<sub>0</sub></b></td><td>&mu;<sub>0</sub></td><td>&sigma;<sub>1</sub></td><td>&mu;<sub>1</sub></td><td>&sigma;<sub>0</sub></td><td>&rho;<sub>0</sub></td><td>&rho;<sub>2</sub></td><td>&rho;<sub>3</sub></td><td>&rho;<sub>1</sub></td></tr><tr align="CENTER"><td><b>&mu;<sub>1</sub></b></td><td>&mu;<sub>1</sub></td><td>&sigma;<sub>0</sub></td><td>&mu;<sub>0</sub></td><td>&sigma;<sub>1</sub></td><td>&rho;<sub>2</sub></td><td>&rho;<sub>0</sub></td><td>&rho;<sub>1</sub></td><td>&rho;<sub>3</sub></td></tr><tr align="CENTER"><td><b>&sigma;<sub>0</sub></b></td><td>&sigma;<sub>0</sub></td><td>&mu;<sub>0</sub></td><td>&sigma;<sub>1</sub></td><td>&mu;<sub>1</sub></td><td>&rho;<sub>1</sub></td><td>&rho;<sub>3</sub></td><td>&rho;<sub>0</sub></td><td>&rho;<sub>2</sub></td></tr><tr align="CENTER"><td><b>&sigma;<sub>1</sub></b></td><td>&sigma;<sub>1</sub></td><td>&mu;<sub>1</sub></td><td>&sigma;<sub>0</sub></td><td>&mu;<sub>0</sub></td><td>&rho;<sub>3</sub></td><td>&rho;<sub>1</sub></td><td>&rho;<sub>2</sub></td><td>&rho;<sub>0</sub></td></tr></tbody></table><P>A group is defined by the following three axioms: </P><ul><li>The binary operation in the group is <i>associative</i>. That is, for all <i>a</i>, <i>b</i>, and <i>c</i> in the group:</li></ul><blockquote><blockquote>(<i>a</i> * <i>b</i>) * <i>c</i> = <i>a</i> * (<i>b</i> * <i>c</i>) </blockquote></blockquote><ul><li>The group contains an <i>identity element</i>. There exists an element <i>e</i> in the group such that for all <i>a</i> in the group:</li></ul><blockquote><blockquote><i>e</i> * <i>a</i> = <i>a</i> * <i>e</i> = <i>a</i></blockquote></blockquote><ul><li>Every element of the group has an <i>inverse</i>. For all <i>a</i> in the group, there exists an element <i>a</i><sup>-1</sup> in the group such that:</li></ul><blockquote><blockquote><i>a</i> * <i>a</i><sup>-1</sup> = <i>a</i><sup>-1</sup> * <i>a</i> = <i>e</i></blockquote></blockquote><P>Associativity is tedious to check for <b>D<sub>4</sub></b>. Associativity implies that one can drop parenthesis below. ρ<sub>0</sub> is the identity element. Every row and column in the multiplication table for <b>D<sub>4</sub></b> contains ρ<sub>0</sub>; thus, every element has an inverse. </P><P>An <i><a href="https://en.wikipedia.org/wiki/Niels_Henrik_Abel">Abelian</a> group</i> is one in which the binary operation is commutative. The group of symmetries of the square is not Abelian. For an Abelian group, the multiplication table is symmetric across the principal diagonal; it does not matter to the result in which order one performs the operation for two arguments. The following two equations illustrates that <b>D<sub>4</sub></b> is not Abelian: </P><blockquote>&mu;<sub>0</sub>*&rho;<sub>1</sub> = &sigma;<sub>1</sub></blockquote><blockquote>&rho;<sub>1</sub>*&mu;<sub>0</sub> = &sigma;<sub>0</sub></blockquote><P>In words, flipping a square around its horizontal axis of symmetry and then rotating it ninety degrees clockwise is not equivalent to rotating it ninety degrees clockwise and then then reflecting it across that axis. The result of the first composition of operations is equivalent to reflecting the square across the diagonal axis of symmetry running from the south west to the north east. The second composition of operations is equivalent to flipping the square across the other diagonal. </P><P>One can also set up equations in a group, for example: </P><blockquote>&rho;<sub>1</sub>*&rho;<sub>2</sub>*<i>x</i> = &mu;<sub>0</sub></blockquote><P>Then <i>x</i> must be &sigma;<sub>0</sub>. Solving a Rubik's cube is analogous to solving such an equation. </P><b>3.0 Proper and Improper Subgroups</b><P>Some rows and columns in Table 1 can stand alone as a group. The entries in these restricted row and columns all appear as headings in the rows and columns. These entries form a <I>subgroup</I> of the original group. One-fourth of the table in the upper left of Table 1 provides an example. {&rho;<SUB>0</SUB>, &rho;<SUB>1</SUB>, &rho;<SUB>2</SUB>, &rho;<SUB>3</SUB>} is a subgroup of <B>D<SUB>4</SUB></B> (Table 2). </P><table align="CENTER" border=""><CAPTION><b>Table 2: A Subgroup of D<sub>4</sub> with Four Elements</b></CAPTION><tr align="CENTER"><td><b>*</b></td><td><b>&rho;<sub>0</sub></b></td><td><b>&rho;<sub>1</sub></b></td><td><b>&rho;<sub>2</sub></b></td><td><b>&rho;<sub>3</sub></b></td></tr><tr align="CENTER"><td><b>&rho;<sub>0</sub></b></td><td>&rho;<sub>0</sub></td><td>&rho;<sub>1</sub></td><td>&rho;<sub>2</sub></td><td>&rho;<sub>3</sub></td></tr><tr align="CENTER"><td><b>&rho;<sub>1</sub></b></td><td>&rho;<sub>1</sub></td><td>&rho;<sub>2</sub></td><td>&rho;<sub>3</sub></td><td>&rho;<sub>0</sub></td></tr><tr align="CENTER"><td><b>&rho;<sub>2</sub></b></td><td>&rho;<sub>2</sub></td><td>&rho;<sub>3</sub></td><td>&rho;<sub>0</sub></td><td>&rho;<sub>1</sub></td></tr><tr align="CENTER"><td><b>&rho;<sub>3</sub></b></td><td>&rho;<sub>3</sub></td><td>&rho;<sub>0</sub></td><td>&rho;<sub>1</sub></td><td>&rho;<sub>2</sub></td></tr></table><P>The group <B>D<SUB>4</SUB></B> has ten subgroups, as shown in the <I>Lattice Diagram</I> in Figure 1 above. Subgroups have been defined such that, for any group <B>G</B>, the group <B>G</B> is a subgroup of itself. Another trivial case, the one-element group consisting of the identity element, also provides a subgroup of <B>G</B>. These two subgroups are known as <I>improper subgroups</I>. All other subgroups are <I>proper subgroups</I>. </P><P>One can make a couple of observations about subgroups. The binary operation in the group is the same as the binary operation in the subgroup. The property of associativity carries over from the group to the subgroup. Since a subgroup is a group, it must contain an identity element. And that identity element must also be the identity element for the group containing the subgroup. Thus, every subgroup of <B>D<SUB>4</SUB></B> contains &rho;<SUB>0</SUB>. Likewise, for every element of a subgroup, the subgroup must also contain its inverse. Finally, the number of elements in a subgroup must evenly divide the number of elements in the group. </P><P>I have shown above how the eight elements of <B>D<SUB>4</SUB></B> can be defined in terms of permutations. As a matter of fact, the set of permutations of (1, 2, ..., <I>n</I>) form a group under the operation of function composition. This <I>permutation group</I> is designated as <B>S<SUB><I>n</I></SUB></B>, and it contains <I>n</I>! elements. Thus, <B>S<SUB>4</SUB></B> contains 24 (= 4x3x2x1) elements. Not only can one find all the subgroups of <B>D<SUB>4</SUB></B>, one can extend the group such that <B>D<SUB>4</SUB></B> is a subgroup of that extended group. </P><b>4.0 Isomorphic Groups</b><P>In a group, the order of rows and columns in the multiplication table are of no matter. Likewise, the names of the elements are irrelevant to the structure of the group. Two groups are <I>isomorphic</I> if the multiplication table for one group can be mapped into the multiplication table for another group by reordering and renaming the elements of, say, the first group. As an example, consider the groups {&rho;<SUB>0</SUB>, &rho;<SUB>2</SUB>, &mu;<SUB>0</SUB>, &mu;<SUB>1</SUB>} and {&rho;<SUB>0</SUB>, &rho;<SUB>2</SUB>, &sigma;<SUB>0</SUB>, &sigma;<SUB>1</SUB>}. They each have the same number of elements, which is necessary for an isomorphism. Table 3 defines the group operation for the first group. Suppose that, in Table 3, &mu;<sub>0</sub> is renamed &sigma;<sub>0</sub>, and &mu;<sub>1</sub> is renamed &sigma;<sub>1</sub> throughout. The resulting table will match the operation for the second group. Thus, the two groups are isomorphic. </P><table align="CENTER" border=""><CAPTION><b>Table 3: The Group {&rho;<SUB>0</SUB>, &rho;<SUB>2</SUB>, &mu;<SUB>0</SUB>, &mu;<SUB>1</SUB>}</b></CAPTION><tr align="CENTER"><td><b>*</b></td><td><b>&rho;<sub>0</sub></b></td><td><b>&rho;<sub>2</sub></b></td><td><b>&mu;<sub>0</sub></b></td><td><b>&mu;<sub>1</sub></b></td></tr><tr align="CENTER"><td><b>&rho;<sub>0</sub></b></td><td>&rho;<sub>0</sub></td><td>&rho;<sub>2</sub></td><td>&mu;<sub>0</sub></td><td>&mu;<sub>1</sub></td></tr><tr align="CENTER"><td><b>&rho;<sub>2</sub></b></td><td>&rho;<sub>2</sub></td><td>&rho;<sub>0</sub></td><td>&mu;<sub>1</sub></td><td>&mu;<sub>0</sub></td></tr><tr align="CENTER"><td><b>&mu;<sub>0</sub></b></td><td>&mu;<sub>0</sub></td><td>&mu;<sub>1</sub></td><td>&rho;<sub>0</sub></td><td>&rho;<sub>2</sub></td></tr><tr align="CENTER"><td><b>&mu;<sub>1</sub></b></td><td>&mu;<sub>1</sub></td><td>&mu;<sub>0</sub></td><td>&rho;<sub>2</sub></td><td>&rho;<sub>0</sub></td></tr></table><P>The groups in Tables 2 and 3 are NOT isomorphic. They each contain four elements. Each element, however, in the group in Table 3 is its own inverse. This is an algebraic property, preserved no matter how the elements of the group are renamed. And the group in Table 2 does not have this property. As a matter of fact, only two groups containing four elements exist, up to an isomorphism. In other words, any group with four elements is isomorphic to either the group in Table 2 or to the group in Table 3. </P><P>Furthermore, only one group, up to isomorphism, contains two elements. Its operation is defined by Table 4. All the subgroups of <B>D<SUB>4</SUB></B> containing two elements are isomorphic to this group and, ipso facto, to each other. The text colors of the subgroups in the lattice diagram (Figure 1) express these isomorphisms. </P><table align="CENTER" border=""><CAPTION><b>Table 4: The Unique Group (Up To Isomorphism) With Two Elements</b></CAPTION><tr align="CENTER"><td><b>*</b></td><td><b>0</b></td><td><b>1</b></td></tr><tr align="CENTER"><td><b>0</b></td><td>0</td><td>1</td></tr><tr align="CENTER"><td><b>1</b></td><td>1</td><td>0</td></tr></table><b>5.0 Normal Subgroups, Factor Groups, and Homomorphisms</b><P>Certain additional patterns are apparent in Table 1. I have already pointed out that the first four rows and columns constitute the subgroup with the operation shown in Table 2. Notice that none of the entries in the last four columns for the first four rows are in this subgroup. Likewise, none of the entries in the first four columns for the last four rows are in this subgroup. On the other hand, the entries in the remaining rows and columns in the lower right are all in this subgroup. Can you see that these observations reveal the pattern expressed in Table 4? Mathematicians express this by saying that the <I>factor group</I> <B>D<SUB>4</SUB></B>/{&rho;<SUB>0</SUB>, &rho;<SUB>1</SUB>, &rho;<SUB>2</SUB>, &rho;<SUB>3</SUB>} is isomorphic to the group with two elements. </P><P>A subgroup is <I>normal</I> if it can be used to divide up the rows and columns in the multiplication table for the group like this. For another example, consider the subgroup {&rho;<SUB>0</SUB>, &rho;<SUB>2</SUB>}. Table 5 shows a reordering of the rows and columns in Table 1 to facilitate the calculation of the factor group for this subgroup. Consider dividing this grid up into 16 blocks of two rows and two columns each. Each block will contain two elements of the group <B>D<sub>4</sub></B>, and which element is paired with each element does not vary among these blocks. </P><table align="CENTER" border=""><CAPTION><b>Table 5: The Group D<sub>4</sub> Reordered</B></CAPTION><tr align="CENTER"><td><b>*</b></td><td><b>&rho;<sub>0</sub></b></td><td><b>&rho;<sub>2</sub></b></td><td><b>&rho;<sub>1</sub></b></td><td><b>&rho;<sub>3</sub></b></td><td><b>&mu;<sub>0</sub></b></td><td><b>&mu;<sub>1</sub></b></td><td><b>&sigma;<sub>0</sub></b></td><td><b>&sigma;<sub>1</sub></b></td></tr><tr align="CENTER"><td><b>&rho;<sub>0</sub></b></td><td>&rho;<sub>0</sub></td><td>&rho;<sub>2</sub></td><td>&rho;<sub>1</sub></td><td>&rho;<sub>3</sub></td><td>&mu;<sub>0</sub></td><td>&mu;<sub>1</sub></td><td>&sigma;<sub>0</sub></td><td>&sigma;<sub>1</sub></td></tr><tr align="CENTER"><td><b>&rho;<sub>2</sub></b></td><td>&rho;<sub>2</sub></td><td>&rho;<sub>0</sub></td><td>&rho;<sub>3</sub></td><td>&rho;<sub>1</sub></td><td>&mu;<sub>1</sub></td><td>&mu;<sub>0</sub></td><td>&sigma;<sub>1</sub></td><td>&sigma;<sub>0</sub></td></tr><tr align="CENTER"><td><b>&rho;<sub>1</sub></b></td><td>&rho;<sub>1</sub></td><td>&rho;<sub>3</sub></td><td>&rho;<sub>2</sub></td><td>&rho;<sub>0</sub></td><td>&sigma;<sub>0</sub></td><td>&sigma;<sub>1</sub></td><td>&mu;<sub>1</sub></td><td>&mu;<sub>0</sub></td></tr><tr align="CENTER"><td><b>&rho;<sub>3</sub></b></td><td>&rho;<sub>3</sub></td><td>&rho;<sub>1</sub></td><td>&rho;<sub>0</sub></td><td>&rho;<sub>2</sub></td><td>&sigma;<sub>1</sub></td><td>&sigma;<sub>0</sub></td><td>&mu;<sub>0</sub></td><td>&mu;<sub>1</sub></td></tr><tr align="CENTER"><td><b>&mu;<sub>0</sub></b></td><td>&mu;<sub>0</sub></td><td>&mu;<sub>1</sub></td><td>&sigma;<sub>1</sub></td><td>&sigma;<sub>0</sub></td><td>&rho;<sub>0</sub></td><td>&rho;<sub>2</sub></td><td>&rho;<sub>3</sub></td><td>&rho;<sub>1</sub></td></tr><tr align="CENTER"><td><b>&mu;<sub>1</sub></b></td><td>&mu;<sub>1</sub></td><td>&mu;<sub>0</sub></td><td>&sigma;<sub>0</sub></td><td>&sigma;<sub>1</sub></td><td>&rho;<sub>2</sub></td><td>&rho;<sub>0</sub></td><td>&rho;<sub>1</sub></td><td>&rho;<sub>3</sub></td></tr><tr align="CENTER"><td><b>&sigma;<sub>0</sub></b></td><td>&sigma;<sub>0</sub></td><td>&sigma;<sub>1</sub></td><td>&mu;<sub>0</sub></td><td>&mu;<sub>1</sub></td><td>&rho;<sub>1</sub></td><td>&rho;<sub>3</sub></td><td>&rho;<sub>0</sub></td><td>&rho;<sub>2</sub></td></tr><tr align="CENTER"><td><b>&sigma;<sub>1</sub></b></td><td>&sigma;<sub>1</sub></td><td>&sigma;<sub>0</sub></td><td>&mu;<sub>1</sub></td><td>&mu;<sub>0</sub></td><td>&rho;<sub>3</sub></td><td>&rho;<sub>1</sub></td><td>&rho;<sub>2</sub></td><td>&rho;<sub>0</sub></td></tr></table><P>These observations can be formalized by the function defined in Table 6. For an element <I>a</I> of <B>D<SUB>4</SUB></B>, let <I>f</I>(<I>a</I>) denote the map defined in Table 6. To find the value of this function, locate <I>a</I> in the first column. Whether this value is 0, 1, 2, or 3 is determined by the corresponding entry in the second column. For all <I>a</I> and <I>b</I> in <B>D<SUB>4</SUB></B>: </P><BLOCKQUOTE><I>f</I>(<I>a</I> * <I>b</I>) = <I>f</I>(<I>a</I>) o <I>f</I>(<I>b</I>) </BLOCKQUOTE><P>A map from one group to another with this property is a <I>homomorphism</I>. An isomorphism is a homomorphism, but a homomorphism is a more general concept. Homomorphisms do not need to leave the number of elements in the group invariant. </P><TABLE ALIGN="CENTER" border=""><CAPTION><b>Table 6: A Homomorphism from D<SUB>4</SUB> to {0, 1, 2, 3}</b></CAPTION><TR ALIGN="CENTER"><TD><B>Elements of D<SUB>4</SUB></B></TD><TD><B>Image</B></TD></TR><TR ALIGN="CENTER"><TD>&rho;<SUB>0</SUB>, &rho;<SUB>2</SUB></TD><TD>0</TD></TR><TR ALIGN="CENTER"><TD>&rho;<SUB>1</SUB>, &rho;<SUB>3</SUB></TD><TD>1</TD></TR><TR ALIGN="CENTER"><TD>&mu;<SUB>0</SUB>, &mu;<SUB>1</SUB></TD><TD>2</TD></TR><TR ALIGN="CENTER"><TD>&sigma;<SUB>0</SUB>, &sigma;<SUB>1</SUB></TD><TD>3</TD></TR></TABLE><P>The factor group <B>D<SUB>4</SUB></B>/{&rho;<SUB>0</SUB>, &rho;<SUB>2</SUB>} is easily calculated. Replace each element of <B>D<SUB>4</SUB></B> in Table 5 by its image under the homomorphism in Table 6. Collapse each pair of rows and columns. One ends up with Table 7, where I have renamed the group operation, as above. The factor group <B>D<SUB>4</SUB></B>/{&rho;<SUB>0</SUB>, &rho;<SUB>2</SUB>} is isomorphic to the group with four elements with the operation shown in Table 3 above. The number of elements in a factor group is the quotient of the number of elements in the original group and the number of elements in the subgroup used to form the factor group. </P><table align="CENTER" border=""><CAPTION><b>Table 7: The Factor Group D<SUB>4</SUB>/{&rho;<SUB>0</SUB>, &rho;<SUB>2</SUB>}</b></CAPTION><tr align="CENTER"><td><b>o</b></td><td><b>0</b></td><td><b>1</b></td><td><b>2</b></td><td><b>3</b></td></tr><tr align="CENTER"><td><b>0</b></td><td>0</td><td>1</td><td>2</td><td>3</td></tr><tr align="CENTER"><td><b>1</b></td><td>1</td><td>0</td><td>3</td><td>2</td></tr><tr align="CENTER"><td><b>2</b></td><td>2</td><td>3</td><td>0</td><td>1</td></tr><tr align="CENTER"><td><b>3</b></td><td>3</td><td>2</td><td>1</td><td>0</td></tr></table><P>The two improper subgroups for any group are normal and yield trivial factor groups. The factor group <B>D<SUB>4</SUB></B>/<B>D<SUB>4</SUB></B> is isomorphic to the one-element group whose only member is the identity element. The factor group <B>D<SUB>4</SUB></B>/{&rho;<SUB>0</SUB>} is isomorphic to <B>D<SUB>4</SUB></B>. The factor groups for improper subgroups provide no information about the structure of a group. </P><B>6.0 A Subgroup that is Not Normal</B><P>Not all subgroups are normal. The subgroup {&rho;<SUB>0</SUB>, &mu;<SUB>0</SUB>}, for example, is not a normal subgroup of <B>D<SUB>4</SUB></B>. Table 8 proposes a map from the elements of the group to the first four natural numbers. And Table 9 illustrates another reordering of the rows and columns in Table 1, with the entries replaced by the natural numbers to which they map. If one confines oneself to the first two columns, each pair of rows could be collapsed into one, with the label from the row taken from the map. But this process breaks down for the next two and the last two columns. </P><TABLE ALIGN="CENTER" border=""><CAPTION><b>Table 8: A Map from D<SUB>4</SUB> to {0, 1, 2, 3} that is Not a Homomorphism</b></CAPTION><TR ALIGN="CENTER"><TD><B>Elements of D<SUB>4</SUB></B></TD><TD><B>Image</B></TD></TR><TR ALIGN="CENTER"><TD>&rho;<SUB>0</SUB>, &mu;<SUB>0</SUB></TD><TD>0</TD></TR><TR ALIGN="CENTER"><TD>&rho;<SUB>1</SUB>, &sigma;<SUB>0</SUB></TD><TD>1</TD></TR><TR ALIGN="CENTER"><TD>&rho;<SUB>2</SUB>, &mu;<SUB>1</SUB></TD><TD>2</TD></TR><TR ALIGN="CENTER"><TD>&rho;<SUB>3</SUB>, &sigma;<SUB>1</SUB></TD><TD>3</TD></TR></TABLE><table align="CENTER" border=""><CAPTION><b>Table 9: Another Reodering of The Group D<sub>4</sub></B></CAPTION><tr align="CENTER"><td><b>*</b></td><td><b>&rho;<sub>0</sub></b></td><td><b>&mu;<sub>0</sub></b></td><td><b>&rho;<sub>1</sub></b></td><td><b>&sigma;<sub>0</sub></b></td><td><b>&rho;<sub>2</sub></b></td><td><b>&mu;<sub>1</sub></b></td><td><b>&rho;<sub>3</sub></b></td><td><b>&sigma;<sub>1</sub></b></td></tr><tr align="CENTER"><td><b>&rho;<sub>0</sub></b></td><td>0</td><td>0</td><td>1</td><td>1</td><td>2</td><td>2</td><td>3</td><td>3</td></tr><tr align="CENTER"><td><b>&mu;<sub>0</sub></b></td><td>0</td><td>0</td><td>3</td><td>3</td><td>2</td><td>2</td><td>1</td><td>1</td></tr><tr align="CENTER"><td><b>&rho;<sub>1</sub></b></td><td>1</td><td>1</td><td>2</td><td>2</td><td>3</td><td>3</td><td>0</td><td>0</td></tr><tr align="CENTER"><td><b>&sigma;<sub>0</sub></b></td><td>1</td><td>1</td><td>0</td><td>0</td><td>3</td><td>3</td><td>2</td><td>2</td></tr><tr align="CENTER"><td><b>&rho;<sub>2</sub></b></td><td>2</td><td>2</td><td>3</td><td>3</td><td>0</td><td>0</td><td>1</td><td>1</td></tr><tr align="CENTER"><td><b>&mu;<sub>1</sub></b></td><td>2</td><td>2</td><td>1</td><td>1</td><td>0</td><td>0</td><td>3</td><td>3</td></tr><tr align="CENTER"><td><b>&rho;<sub>3</sub></b></td><td>3</td><td>3</td><td>0</td><td>0</td><td>1</td><td>1</td><td>2</td><td>2</td></tr><tr align="CENTER"><td><b>&sigma;<sub>1</sub></b></td><td>3</td><td>3</td><td>2</td><td>2</td><td>1</td><td>1</td><td>0</td><td>0</td></tr></table><P>Suppose a subgroup contains <I>n</I> elements. To determine if the subgroup is normal, it is sufficient to examine the first <I>n</I> rows and the first <I>n</I> columns in the reordered table. This capability follows from a theorem about what are known as left and right cosets for a subgroup. </P><P>The permuation group <B>S<SUB>4</SUB></B> provides another example of a subgroup that is not normal. By my calculations, <B>D<SUB>4</SUB></B> is NOT a normal subgroup of <B>S<SUB>4</SUB></B>. </P><b>7.0 The Composition Series of a Group</b><P>At this point, I have completed my explanation of the lattice diagram at the top of this post, including circles, text colors, and boxes. I draw from these results to illustrate how a non-simple group, namely <B>D<SUB>4</SUB></B>, can be expressed as a composition of factor groups. </P><P>Table 10 lists twelve series of subgroups of the group of symmetries of the square. Each series has the following properties: </P><UL><LI>The leftmost group in the series is the one-element group containing the identity element.</LI><LI>The rightmost group is <B>D<SUB>4</SUB></B>.</LI><LI>Each group in the series (except <B>D<SUB>4</SUB></B>) is a proper normal subgroup of the group immediately to the right of it in the series.</LI></UL><P>A series with these properties is known as a <I>subnormal series of the group</I> <B>D<SUB>4</SUB></B>. If every group in the series is also a normal subgroup of <B>D<SUB>4</SUB></B>, the series is a <I>normal series of the group</I> <B>D<SUB>4</SUB></B>. By the last property in the bulleted list, one can calculate a factor group for each pair of immediately successive groups in the series. </P><TABLE align="CENTER" border=""><CAPTION><B>Table 10: Twelve Normal and Subnormal Series for <B>D<SUB>4</SUB></B></B></CAPTION><TR align="CENTER"><TD><B>Number<BR>Factor Groups</B></TD><TD><B>Series</B></TD><TD><B>Normal<BR>Series</B></TD></TR><TR align="CENTER"><TD>1</TD><TD>{&rho;<SUB>0</SUB>} &lt; <B>D<SUB>4</SUB></B></TD><TD>Yes</TD></TR><TR align="CENTER"><TD>2</TD><TD>{&rho;<SUB>0</SUB>} &lt; {&rho;<SUB>0</SUB>, &rho;<SUB>1</SUB>, &rho;<SUB>2</SUB>, &rho;<SUB>3</SUB>} &lt; <B>D<SUB>4</SUB></B></TD><TD>Yes</TD></TR><TR align="CENTER"><TD ROWSPAN="3">2</TD><TD>{&rho;<SUB>0</SUB>} &lt; {&rho;<SUB>0</SUB>, &rho;<SUB>2</SUB>, &mu;<SUB>0</SUB>, &mu;<SUB>1</SUB>} &lt; <B>D<SUB>4</SUB></B></TD><TD>Yes</TD></TR><TR align="CENTER"><TD>{&rho;<SUB>0</SUB>} &lt; {&rho;<SUB>0</SUB>, &rho;<SUB>2</SUB>, &sigma;<SUB>0</SUB>, &sigma;<SUB>1</SUB>} &lt; <B>D<SUB>4</SUB></B></TD><TD>Yes</TD></TR><TR align="CENTER"><TD>{&rho;<SUB>0</SUB>} &lt; {&rho;<SUB>0</SUB>, &rho;<SUB>2</SUB>} &lt; <B>D<SUB>4</SUB></B></TD><TD>Yes</TD></TR><TR align="CENTER"><TD ROWSPAN="7">3</TD><TD>{&rho;<SUB>0</SUB>} &lt; {&rho;<SUB>0</SUB>, &rho;<SUB>2</SUB>} &lt; {&rho;<SUB>0</SUB>, &rho;<SUB>1</SUB>, &rho;<SUB>2</SUB>, &rho;<SUB>3</SUB>} &lt; <B>D<SUB>4</SUB></B></TD><TD>Yes</TD></TR><TR align="CENTER"><TD>{&rho;<SUB>0</SUB>} &lt; {&rho;<SUB>0</SUB>, &rho;<SUB>2</SUB>} &lt; {&rho;<SUB>0</SUB>, &rho;<SUB>2</SUB>, &mu;<SUB>0</SUB>, &mu;<SUB>1</SUB>} &lt; <B>D<SUB>4</SUB></B></TD><TD>Yes</TD></TR><TR align="CENTER"><TD>{&rho;<SUB>0</SUB>} &lt; {&rho;<SUB>0</SUB>, &rho;<SUB>2</SUB>} &lt; {&rho;<SUB>0</SUB>, &rho;<SUB>2</SUB>, &sigma;<SUB>0</SUB>, &sigma;<SUB>1</SUB>} &lt; <B>D<SUB>4</SUB></B></TD><TD>Yes</TD></TR><TR align="CENTER"><TD>{&rho;<SUB>0</SUB>} &lt; {&rho;<SUB>0</SUB>, &mu;<SUB>0</SUB>} &lt; {&rho;<SUB>0</SUB>, &rho;<SUB>2</SUB>, &mu;<SUB>0</SUB>, &mu;<SUB>1</SUB>} &lt; <B>D<SUB>4</SUB></B></TD><TD>No</TD></TR><TR align="CENTER"><TD>{&rho;<SUB>0</SUB>} &lt; {&rho;<SUB>0</SUB>, &mu;<SUB>1</SUB>} &lt; {&rho;<SUB>0</SUB>, &rho;<SUB>2</SUB>, &mu;<SUB>0</SUB>, &mu;<SUB>1</SUB>} &lt; <B>D<SUB>4</SUB></B></TD><TD>No</TD></TR><TR align="CENTER"><TD>{&rho;<SUB>0</SUB>} &lt; {&rho;<SUB>0</SUB>, &sigma;<SUB>0</SUB>} &lt; {&rho;<SUB>0</SUB>, &rho;<SUB>2</SUB>, &sigma;<SUB>0</SUB>, &sigma;<SUB>1</SUB>} &lt; <B>D<SUB>4</SUB></B></TD><TD>No</TD></TR><TR align="CENTER"><TD>{&rho;<SUB>0</SUB>} &lt; {&rho;<SUB>0</SUB>, &sigma;<SUB>1</SUB>} &lt; {&rho;<SUB>0</SUB>, &rho;<SUB>2</SUB>, &sigma;<SUB>0</SUB>, &sigma;<SUB>1</SUB>} &lt; <B>D<SUB>4</SUB></B></TD><TD>No</TD></TR></TABLE><P>The definition of an isomorphism for a subnormal series builds on the definition of isomorphism for groups. Consider the factor groups arising in each series from successive pairs of subgroups in each series. Two series are isomorphic if they contain the same of number of factor groups, in this sense, and these factor groups are isomorphic. The order in which the factor groups arise can vary among isomorphic subnormal series. </P><P>I have collected isomorphic series together, in Table 10, by means of horizontal lines in the first column. The series with one factor group is not isomorphic to any other series. The first series shown with two factor groups is not isomorphic to the other three series with two factor groups. And those three series are isomorphic to one another. All of the series with three factor groups are isomorphic to one another. </P><P>The series with three factor groups have another property. All factor groups in these series with three factor groups are simple groups. That is, they contain no proper normal subgroups. A subnormal series of a group in which all factor groups formed by the series are simple is known as a <I>composition series</I>. By the Jordan-H&ouml;lder Theorem, all compositions series for a group are isomorphic series. This theorem justifies one in speaking of THE composition series for a group. Finding the factor groups in a the composition series for a group is somewhat analogous to factoring a natural number. Note that <B>D<SUB>4</SUB></B> contains eight elements and each of the three factor groups in the composition series contain two elements. Furthermore, </P><BLOCKQUOTE>8 = 2<SUP>3</SUP></BLOCKQUOTE><P>For a natural number, the prime factors can be combined to yield the original number. Here the analogy apparently breaks down. The factor groups in a composition series for a group constrain the structure of the group, but two non-isomorphic groups can have the same composition series. But still, mathematicians have solved various problems in group theory for finite non-simple groups by use of the classification of finite simple groups. </P><P>Composition series apparently have an application in solving polynomial equations. The composition series for the permutation group <B>S<SUB>5</SUB></B> contains a factor group that is non-Abelian. This is connected with the insolvability of the quintic. There are formulas for zeros for cubic and fourth order polynomial, analogous to the quadratic formula. But there is no such formulas for poynomials of the fifth degree and higher. </P><b>8.0 Classification of Finite Simple Groups</b><P>At this point, I have explained how finite simple groups arise as factor groups for the composition series of any finite group. I hope that this gives some hint of why the following theorem is of interest. </P><P><B>Theorem:</B> Each finite simple group is one of the following, up to an isomorphism: </P><UL><LI>A group of prime order.</LI><LI>An alternating group.</LI><LI>A Lie group.</LI><LI>One of 26 sporadic groups not otherwise classified.</LI></UL><P>I am aware that this this theorem uses technical terms I still have not explained, including one that I simply do not understand myself. </P><P>The sporadic groups are finite simple groups that do not fall into the other categories, although, I gather, some sporadic groups are related to one another.The sporadic group with the largest number of elements is called the Monster group. It has 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 elements. </P><B>9.0 History of the Theorem</B><P>In 1972, Daniel Gorenstein proposed that mathematicians could complete a classification of all simple groups. By the early 1980s, mathematicians had stated the theorem and those specialists who had pursued Gorenstein's program believed they had proven it. The proof, however, was scattered among (tens of?) thousands of pages in hundreds(?) of papers in many mathematics journals. No one person had probably ever understood the proof or read it in its entirety. </P><P>The proof, however, was discovered even then to be incomplete. Steve Smith and Michael Aschbacher worked on closing this gap, relating to <I>quasithin</I> groups. They succeeded by 2004. </P><P>Meanwhile, a number of mathematicians have been trying to simplify the proof and to restate it in one location. The ambition of these mathematicians is to produce a "second generation" proof of only a couple thousand pages. </P><P>Has a theorem been proven if only one or two mathematicians have read the proof in its entirety? How about if nobody has, which would have been the case in the 1980s if the proof had indeed been valid? Certainly, the proof of the classification theorem is not surveyable, in Wittgenstein's sense. Do mathematical results need to be established by a social process? If so, how can such social processes be characterized? </P><b>Appendix: Terms Defined or Illustrated Above</b><P>Abelian group, Associativity, Composition Series, Factor Group, Finite Group, Group, Homomorphism, Identity Element, Improper Subgroup, Inverse, Isomorphic Groups, Isomorphic Subnormal Series, Lattice Diagram, Normal Series, Normal Subgroup, Permutation Group, Proper Subgroup, Subgroup, Subnormal Series. </P><b>References</b><ul><LI>Michael Aschbacher (2004). The Status of the Classification of the Finite Simple Groups, <I>Notices of the AMS</I>, V. 51, No. 7 (Aug.): pp. 736-740.</LI><li>Michael Aschbacher, Richard Lyons, Stephen D. Smith, and Ronald Solomon (2011). <a href="http://www.amazon.com/Classification-Finite-Simple-Groups-Characteristic/dp/0821853368/"><i>The Classification of Finite Simple Groups: Groups of Characteristic 2 Type</i></a>, American Mathematical Society.</li><li>Nicolas Bourbaki (1943). <i>Elements of Mathematics: Algebra I: Chapters 1-3</i>.</li><LI>J. H. Conway and S. P. Norton (1979). <A HREF="http://blms.oxfordjournals.org/content/11/3/308.extract">Monstrous Moonshine</A>, <I>Bulletin of the London Mathematical Society</I>, V. 11, no. 3: pp. 308-339.</LI><li>John B. Fraleigh (2002). <i>A First Course in Abstract Algebra</i>, 7th Edition, Pearson.</li><li>Daniel Gorenstein, Richard Lyons, and Ronald Solomon (1994). <a href="http://www.amazon.com/gp/product/0821809601/ref=pd_lpo_sbs_dp_ss_3?pf_rd_p=1944687742"><i>The Classification of the Finite Simple Groups</i></a>, American Mathematical Society.</li><LI>Ian Hacking (2014). <I>Why is there Philosophy of Mathematics at all?</I>, Cambridge University Press.</LI><LI>Daniel Kunkle and Gene Cooperman (2007). Twenty-Six Moves Suffice for Rubik's Cube, <I>ISSAC'07</I>, 29 Jul. - 1 Aug., Waterloo, Canada.</LI><LI>Tomas Rokicki (2008). Twenty Five Moves Suffice for Rubik's Cube.</LI></ul>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com2tag:blogger.com,1999:blog-26706564.post-64730449728154567612016-02-11T14:51:00.000-05:002016-02-11T14:51:04.826-05:00European Monetary Union Without Political Union<P>I recently read Richard Davenport-Hines' <A HREF="http://www.amazon.com/Universal-Man-Lives-Maynard-Keynes/dp/0465060676"><I>Universal Man: The Lives of John Maynard Keynes</I></A>. One thing I learned was of the existence of the <A HREF="https://en.wikipedia.org/wiki/Latin_Monetary_Union">Latin Monetary Union</A>. </P><P>Apparently, in the latter half of the nineteenth century, gold and silver coins circulated in a number of European countries in which they speak Romance languages. And the amount of gold or silver in these coins was specified. I guess this is part of being on the gold standard. I gather the countries in the Latin Monetary Union agreed on a fixed ratio of silver to gold. As part of this agreement, coins from all these countries circulated freely throughout these countries. You could spend a franc coin in Italy just as conveniently as a lira coin. </P><P>I am surprised that this union lasted past World War I. From Keynes' <A HREF="http://delong.typepad.com/keynes-1923-a-tract-on-monetary-reform.pdf"><I>Tract on Monetary Reform</I></A> (1924), I recall something about the European inflations and deflations that hit Europe after World War I. Yet from my limited reading, I do not recall much about the stresses that must have arisen in this monetary union. Larger issues seem to me to revolve around how the allies in the United States in the war could pay off their loans and how Germany could pay their reparations, agreed to at Versailles, while abiding by the limitations on their economy - such as the occupation of the Ruhr - imposed by the allies. My interest here might be biased by my interest in Keynes, since these issues were a major point of <I>Economic Consequences of the Peace</I>. </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com2tag:blogger.com,1999:blog-26706564.post-45941549291139715282016-01-23T11:39:00.000-05:002016-01-25T07:13:52.285-05:00Two Views On Introductory Economics<P>Recently, two bloggers have commented on what is taught in college classes for introduction to economics<SUP>1</SUP>. Noah Smith <A HREF="http://noahpinionblog.blogspot.com/2016/01/101ism.html">accepts</A> simple partial equilibrium models of perfect competition as internally valid<SUP>2</SUP>. He argues, however, that "Economics 101" models should be complemented, especially in policy applications, with complications introduced in more advanced models. Robert Paul Wolff, on the other hand, <A HREF="http://robertpaulwolff.blogspot.com/2016/01/i-yield-to-thundering-demand.html">uses</A> <A HREF="http://robertpaulwolff.blogspot.com/2016/01/i-start-to-respond-to-comments.html">introductory</A> <A HREF="http://robertpaulwolff.blogspot.com/2016/01/more-responses-to-comments.html">economics</A> as an <A HREF="http://robertpaulwolff.blogspot.com/2016/01/a-lengthy-response-to-wallace-stevens.html">example</A> of ideological bullshit, to use Frankfort's technical term. </P><P>As far as I am concerned, simplistic supply-and-demand reasoning has been shown to be an incoherent mishmash decades ago. Like Prof. Wolff, I like to justify this view by referring to accepted findings of research literature. I particularly like to emphasize the supposed <A HREF="http://robertvienneau.blogspot.com/2006/12/wages-and-employment-not-determined-by.html">market</A> for <A HREF="http://robertvienneau.blogspot.com/search/label/Labor%20Markets">labor</A>. Why do economists not revise their teaching<SUP>3</SUP> so it is not susceptible to being criticized as ideology? I offer three suggestions to complement Wolff's treatment. </P><P>First, perhaps economists who teach outdated nonsense are just doing their job. Introductory courses are followed by later courses. And teachers of later courses expect students who have satisfied the prerequisites to have been exposed to graphs of supply and demand functions, the theory of utility maximization, marginal cost, marginal revenue, the First Order Conditions for maximization, consumer and producer surplus, etc. You might hope for teachers who introduce a bit of pluralism. But even economists who agree with me might find it challenging for the students to be both exposed to critiques and alternatives, and yet gain a command over the conventional material. </P><P>Second, perhaps the situation might be thought of as a type of coordination game, as in modeling a totalitarian society. Maybe the majority of economists privately think that they are being asked to teach balderdash. But, with the profession being the way it is, they see little benefit in saying so. Each sees others as publicly accepting what is being taught. So they put their doubts aside. If all were to be forthright at once, the situation would be different. But how could teaching transverse from the current equilibrium to that new one? </P><P>Third, maybe many economists come to accept what they are teaching as a way of managing cognitive dissonance. It must be an uncomfortable feeling to know one is spouting nonsense and, if one wants to advance in the profession, to be impotent to change it. Better come to accept the nonsense<SUP>4</SUP>. </P><B>Footnotes</B><OL><LI>Both bloggers seem to be concentrating on microeconomics.</LI><LI>Is Noah's conflation of <I>elasticity</I> with the slope of a function an acceptable simplification for a mass audience? Or just muddle?</LI><LI>I do not teach.</LI><LI>I guess this is related to the <A HREF="https://en.wikipedia.org/wiki/Just-world_hypothesis">just world fallacy</A>.</LI></OL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com1